A PROBABILITY MODEL FOR 'FHEORY OF ORGANIZATION OF GROUPS WITH MULTI . VALUED RELATIONS BETWEEN PERSONS Thesis for the Degree OI DI'I. D. MICHIGAN STATE UNIVERSITY John Lucian Bagg 1956 LIBRARY Michigan State University This is to certify that the thesis entitled A Probability Model for Theory of Organization of Groups with Multi-Valued Relations Between Persons presented by John Lucian Bagg has been accepted towards fulfillment of the requirements for Doctor of Philosophy— degree inmathematical Statistics I /, .' - -‘ . 'I-.' ,1 . ’- M19 A ,1.) i, Major profess .6 J Date August 1, 1256 0-169 we... LIBRARIES .——. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. A PROBABILITY mm FOR THEORY OF ORGANIZATION 01' GROUPS WITH MLETI-VALUED BIBLE IONS BETH“ PERSONS ‘ W John Lucian Bags All ABSTRACT Submitted to the School of Advanced Graduate Studies of Michigan state University in Partial rulfillmont of the Requirements for the Degree of WOTOB OF PHILOSOPHY Department of statistics Year 1956 Q \-\ \ -\ \. J OHN LUCIAN BAGG ABSTRACT This thesis is concerned with a probability model for youp organisation theory permitting multi-valued relations between persons. The most difficult problun in connection with this model was the development of a method of enunerating the elements in the universe or discourse. The introduction briefly discusses the development of the matrix model presently being used in connection with sociometric tests. In addition the evolution of the extension to multi-valued relations is examined and a comparison is made between the difficulties surmounted in this study and those in the previous original study for the binary case. The beginning portion of the next section sets up the machinery to be used in the three stage developnent leading to the main theorem. Due to the complexity of the problem it was considered important to indicate the evolution of the process which led to the general result. The results of the first two stages are special cases of the main theorem. The main theorem provides a method for enumerating the total nunber of ways in which n persons can classify seen of the other (n-l) persons into one and only one of (k+ 1) categories. with no restrictions on the number in any given category. This number is given in two wave. first as an expression involving untabulated hollow bipartitional functions. and second in terms of regular J ‘4.- w— “. - -'-'~ .-e~_;n_x:.. . ‘ ..- r:...‘— -"’-‘—'- ~- A‘- — a“; J'— bipartitional functions tabulated by David and Kendall)‘. This theorem makes possible the formulation of exact probability distributions for a large class of random variables. The last part of the thesis indicates a procedure for defining such random variables on a subspace of the sample space of all outcomes. Given a socio- metric test. the nature of the test determines the universe of dis- eourse for these random variables. Once this is decided. the definitiat of the random variable is a matter of expediency. For a large class of sociometrie tests. e.g.. those satisfying restrictions given previously. the exact distribution of these random variables can be found by methods developed in this paper. see reference [l], c" Rvn-“A/ " A PmBABILITY mm FOR THEORY OF ORGANIZATION OF GROUPS WITH MULTI-VALWD RELATIONS 3mm PERSONS By JOHN LUCIAN Em A THESIS Submitted to the school of Advanced.Graduate Studies of Michigan State University in Partial fulfillment of the Requirements for the Degree of NOTOH OF PHILOSOPHY Department of Statistics 1956 I 5/3 ' ,e - f‘ I if: John Lucian Bagg candidate for the degree of Doctor of Philosophy l'inal lamination: August 1 . 1956. 1:15 P.M.. Physics-Mathematics Conference Room Dissertation: A Probability Model for Theory of Organisation of' Groups with Multi-valuod Relations between Persons Outline of Studies Major subject: Mathematical statistics Minor subjects: Algebra, analysis. geometry Biographical Items Born. February 28; 1916. DeKalb. Illinois Undergraduate studies. Northern Illinois State Teachers College. 1934-1938 Graduate Studies. University of Illinois. 1939-1940. Michigan state College and Michigan State University. 1950-1956 Experience: Instructor. Hannibal LaGrange Junior College. 1940-1941; Actuarial Clerk. Metropolitan Life Insurance Company, 1941-1943; Military Service. 1942-1946; Instructor. Keuka College. 1946-1947; Maryland State Teachers College, 1947-1948; Florida State University. 1948-1950; Graduate assistant, Michigan state College. 1950-1951; Temporary Instructor. Michigan state College and Michigan state University. 1951-1956. Member of Signs Zeta. Pi Mu Epsilon; member of Institute of Mathematical Statistics. Merican Mathematical Society. Mathematical Association of Marisa ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. Lee Kate, his major professor. fer suggesting the problem and for his continuous support and patience during the completion of the problem. The writer also deeply appreciates the financial support of the Office ef‘Naval Research which.made it possible for him to continue his work during its early stages. TABLE OF CONTENTS mmmCTIon eeeeeeee O. ..... 0 eeeeeee ooeeeOOeOOO ...... o. 0.1 The developent of the matrix model in sociometric theory ....... . ..... .. . .. ....... . 0.2 Evolution of the problem .................... 0.3 A comparison with a previous study ....... PART 1. TEDDY . ....................................... 1.1 Preliminaries ..... . ................. ........ 1.2 The product space ...................... . 1.3 The operators ....... W. 1.4 The three stages of development .. ..... . 1.5 3(51. :1. £3. 32) as a bilinear form of %§..21u) “a 1“?“ :2“) es eeeeeeeee oeeee 106 H(£1. £1. e00, f. at) u ‘ Mt1.‘ Of «(5%. £1“)eee 1(£ku. é) use eeeeee eeee eeeee PART II. APPLICATIONS .............. . ............. 2.1 Preliminaries ......... 2.2 Baudem variables defined on the matriees c .. 2.3 Some sanple spaces defined by test instruetions ..... . ......................... . 2.4 A hypothetieal emple .. .. ............. .. SW .0. eeeeeeeeeeeeeeee OOOOOOOOOOOOOOOeeOO 000000 O. flmnlx eeeeeeeeee O 0000000000 0.000.... ..... o OOOOOOOOO O A.l Preliminaries ........ ....................... 1.2 Method of attack ................. 1.3 3(1‘. 31. 1‘. 33) as a bilinear form of (1". 31) and (1‘, 32) . ........ . ....... .. RUMORS ..................... . ................. PIC. H (DUIH 10 10 13 15 19 48 51 5? 62 62 68 73 IMMDWTIOH 0:1 The development of the matrix model in sociometric theory. Many investigations. especially in the social sciences. involve the process of having individuals indicate one or more choices from a group of alternatives. The best known of this type of investigation is that referred to in present day literature as sociometry. Soci- onetry. introduced by Moreno [b]. 1. a technique of measuring relo- tionahips in social groups by the simple expedient of asking indi- viduals to indicate preferences for other individuals in the group. is developed by Moreno [6]. the sociometric test placed no rostrictions on the persons within the group who may be chosen or rejected. Further. the subjects were permitted an unlimited nmber of choices or rejections. Modifications have since been introduced by some investigators which limit the umber of choices or rejections made by each individual. Until 1946 the only acceptable method of emu-arising the results of a sociometric test was the sociograa. first described by Moreno [6]. The sociogran consisted of a configuration of directed lines emanating from and terminating on a finite set of points. On the basis of a single activity the members of a group (represented by the n points P1. P2. .... P3) would be asked to respond to the other members. the responses being represented by directed lines between ordered pairs of points. For exasple. with regard to the question ”with when would you like (or definitely not like) to work on a given detail't". if individual i _1_i_n_gg to work with individual j then a directed _s_o_1_i:d_ line from P1 to Pj represented this response: if individual i definitely did get l__ik_e_ to work with individual .1 then a directed dot-dash line from P1 to P3 represented this response: if no response was made then no line was drawn. This method of studying a group with regard to a single activity produced results: however. even thou@ sociogr-is are widely used today there is yet no single convention for the construction of the diagram As a result there are as many different ding-as for a given study so than are investigators.even thougi all are using the one data. In 1946. Forsyth and late [3] presented sociologists with the matrix representation of responses between ordered pairs of individuals in a group on the basis of a single activity. Their representation of the responses was isomorphic to the sociogr-i in that there was a one-to-one correspondence between types of directed lines and types of elements in an nxn matrix C 7- (on). That is. with regard to the question given previously. if individual i 111593 to work with individual j. then on 1' 4- (corresponding to a solid line): if individual i definitely did not L19. to work with individual j. then on = — (corresponding to a dot-dash line); otherwise c1 3 was left blank (corresponding to no line). In this first matrix representation on E x. for all i. and no meaning is attached to the symbol. The may-ix representation has obvious advantages over the socio- gram. both from the point of view of analysis and of the systenatisation ‘- -'—I -_—-l_v.-—'. . ‘h—m‘-‘ -~—-. a:-—-.-.- \. ah?“ _. 7 - ._ of the responses. This becomes very evident when the sire of the group is increased and/or the nunber of types of responses is increased. about a year later tats [4] reformulated the matrix approach to the representation of responses between ordered pairs of individ- uals in a map. It was realised that reactions or responses m be expressed in various weys on different scales. but in any scale there is the possibility of no response at all. No response (usually termed 'indifference') was now given recognition in the nxn matrix C 3 (cu) by on = 0. rurthennore. the symbols + and - used formerly were new replaced by the scalars l and --1 respectively. In this paper by lets the main diagonal elwnents were used as identi- fiers of the individuals in a group. it a later date the convention ‘ii 5 0 was adopted and corresponding row and column indexes identi- fied group manbers. assming the members have been arbitrarily nmbered. Thus. with regard to a single activity of a group. the ternary responses between ordered pairs of individuals were represented by an nxn matrix 0 = (cu). where cu = 1. 0. or -1. The main purpose of this article was to danonstrate that certain matrix opera- tions could be interpreted in a sociological sense. A special case of ternary responses is taken up in the appendix and a more general case in section 1.5 of this paper. At this time no attempt was made to find probability distributions associated with certain classes of these matrices. In order to specify completely such distributions it was necessary to develop ———D---' _ _.——— exact counting procedures. based on the assunption that all matrices in a given class were equally likely. and such procedures were yet to be feund. The first success along this line is given in a paper by Rats and Powell [5]. rreqnent reference will be made to this particular paper. since the main results which follow are extensions of their work. This paper gives the number of locally restricted directed graphs. These graphs consist of t directed lines on n points or nodes; the lines are Joins from one point to another. The graph of t lines on n points or nodes corresponds uniquely to an on matrix C = (cu). That is. c1: = 1. if and only if a directed line connects nods P1 to node P3. otherwise on = 0. Obviously c“ E 0 for all 1. Thus the matrix exhibits the binary relations between ordered pairs of distinct individuals in a group of n individuals. Associated with each node P1 is a non-negative amber pair (r1.s1). where r1 is the number of lines emanating from P1 and s1 is the amber of lines terminating on P1. In the matrix 0 we n then have the ith row total r1 : Ed”. and the 3th colunn total n s: = E on . rurthermore. if the total umber of directed lines is n n E: = S- : : t. then 1.1 r1 3:1 I: t. to let 3.; (r1. r2. .... rn) and s = (s1. s2. .... on) be two n-part. non-negative. ordered partitions of t. thus 5 and _s_ are respectively the marginal row and column totals of the matrix 0. Iith this notation rats and Powell [5] noted that the nusher of i.._ .—-—————-’ .— locally restricted directed graphs. «(5. g). is identical with the nunber of distinct matrices O of seroes and ones. subject to the local restrictions 5 and 3 plus the restriction that °ii E O for all 1. Their main theorem expresses “(3. g) as a linear com- bination of Sukhatme's [8] functions “3“. g“). the latter being the amber of ways in which the cells of a PIC“ lattice (take 9 : 0’ = n) can be filled with seroes and ones so that (l) the row totals from top to bottom form the partition 5 in some fixed order and (2) tho calm totals from left to right form the parts of the partition 3 in some fixed order. Iith this theorem it was possible to find exact probability distributions of random variables defined on the sanple space given by the local restrictions. 0.2 lvelution of Problan It is the purpose of this paper to extend the result of Eats and Powell [5] beyond the binary case. Their counting theorem can be applied only when the responses or reactions between the ordered pairs of distinct individuals in a group. with regard to a single criterion. fall into two categories or can be put on a two point scale. In a large umber of sociometric investigations this is sufficient but it is obviously restrictive. The first success in extending “the results of lots and Powell [6] will be found in the appendix. There a counting theorem is proved for a restricted ternary case of responses. That is. with reference to the question used at the beginning. each person in a group of n individuals must choose exactly one person with whom he would L12 to work and exactly one other with whom he would definitely go; ilk; to work. and the remaining (n-S) individuals are automatically put into the no response category. In the matrix representation of these responses. this implies that each row has exactly one +1 (repre- senting the “like" category). exactly one --1 (representing the “not like“ category). and (n-3) seroes (representing the "no response" category). The corresponding seciegrsm. in this special case. has two lines emanating from each node P1. one solid line (representing the ”like- category). and one dot-dash line (representing the “not like' category). Since the method of proof for this special case is unique it was included in this paper. Also. without doubt. it provided an impetus end an insidit toward the solution of the more general cases. .. The next step was proving a counting theorem when the previous restrictions in the ternary case were relaxed. That is. each indi- vidual was permitted to classify any number of individuals in each of the three categories. subject only to the restriction that each person must classify each of the others into one and only one of the three categories. Theorao 1.5.4 of section 1.5 gives a method of counting the nmber of such matrices under certain other arbitrary but realistic restrictions. Thus. subject to certain practical and realistic restrictions. this theoran will enable sociologists to find exact probability distributions of random variables defined over such sanple spaces as (t1. t2). where t1 and t; are the total number of individuals classified into categories one and two respectively. obviously. in this restricted ternary case. the total number classified in the third category is n-l-tl-tz. Sinilarly. let rm be the amber of individuals classified by individual i in the nth category (u 2 l. 2; i = l. 2. .... n) and form the .. n-dimensional vectors £1 = (r11. r21! ’00. Tu) “d .1} : (1'12. 1'22. eeo. r“). n: m arbitrary but fixed vectors 51 and 53. subject only to the re- strictions r114- r1.2 é.(n-l) for each i. we can now find the exact probability distribution of (31. 33). which are the n-dimeusional vectors representing the number of times each individual was clas- sified into categories one and two respectively. Note that rn + r12 é (n-l). for all i. are necessary but not sufficient condi- tions to insure the realistic restriction that each person classify each of the others into one and only one category. It is admitted that the amber of calculations. even for small groups. would be formidable but. at the one time. they are not impossible. Achieving success in the ternary case the next case considered was not the case of four categories. as might be expected. but the general case of (k + l). 1: >1. categories. from a sociemetric view- point this would cover 931 test satisfying the restriction that each person classify each of the others in one and only one category. For example. tests based on cmplote ranking or partial ranking of the members of a group could be employed. and using the counting theorem. exact probability distributions for certain random variables could be computed. Irhere is a reason for referring to this case as (k+ 1) categories. If the necessary restriction that each masbor of the group classify each of the others into one and only one category is a premise. then any individuals not classified by individual i in one of the first 1: categories would automatically be classified in the (k+ 1)st category. In some instances. depending on the criterion or test. the (k +1)st category will be the no response category. while in other cases this category will be of the one nature as the other 1: categories and there will not be a no re- sponse category. o.g. the case of complete ranking. i'ho counting theoran for this general case of (k+ 1) categories of response with regard to a single activity of a group will be found in section 1.6 0.3 A comparison with a previous study. It is of interest to compare the problems and methods of attack in this paper with those of Kate and Powell [5]. Their main problem. with regard to the counting theorem. was to either find a direct coa- binatorial method of counting the matrices (subject to the local restrictions) or to find a method involving known functions. They accomplished the latter by proving that «(35. p) is a linear combina- tion of Sukhatme's [a] Bipartitional runctions “5“. g“). is was pointed out earlier. the “a“ . 3“) function was concerned with matrices having seroes or ones in am position. including the main diagonal. They accomplished this by first proving that Mg. 5) is a“, -'-c<)' using rellor|s [2] principle of inclusion and exclusion. Then. by taking the inverses of functions a linear combination of [v of certain operators. which they discovered. they were able to express r65. 3 ) as a linear combination of “5“. 30‘) functions. In the first phases of the research in this paper to extend the results of tats and Powell [5] an attempt was made to find a related tabulated function. similar to Sukhatme'l Bipartitional Functions [8], to cope with the ternary responses. This search proved futile. however. and a different line of attack had to be taken. The method finally adopted in the ternary case was the fuming of a product space which contained as a proper subspace the desired elements. The details of this will be found in section 1.2. Briefly described. the method is as follows. Using the methods of Kata and Powell [5] it is possible to form 1: independent sets of nan matrices. one for each of the 1: categories (excluding the (k+ 1)st ). The product space of these k independent sets is then a space of three dimensional. nxnxk. matrices. Since the 1: sets are independent. the resulting product space contains. in general. mar: nxnxk matrices which violate the necessary restriction that each of the n manbers of a group classify each of the others in one and only one of the (k + 1) categories. The preblcu at this stage is then similar to that encountered by Kate and Powell [5].llane1y. a method of counting only those matrices in the product space which do not violate the necessary restriction had to be found. The general counting theorem will be found in section 1.5. 10 PART I THEORY 1:1. Preliminaries. This study is concerned with relations between ordered pairs of distinct individuals in a finite group of n individuals. It is assuned that there are (Ir? 1) distinct types of relations. Is in- sist that one and only one of the (k-t- l) relations exist between eaCh ordered pair (1.3) of distinct individuals (um; i.) = 1. s. . ... n). In most instances the (k-t- 1)st relation will be known as the null relation. However. in some sociometric tests the (k+1)st relation will be of the suns nature as the other 1: relations; in this event it would be treated in the one manner as the null relation. This would occur in sociometric tests in which it is mandatory that every individual be classified into non-null categories. The totality of relations between all ordered pairs of distinct individuals can be represented in two ways which are isomorphic. One way is fmniliar to the social scientist and consists of graphs. The graphs have n points or nodes P1 (i = 1. 2. .... n). Between each pair of points P1 and PJ there is either one and only one of k different colored directed lines or there is no directed line. the absence of a line corresponding to the null relation. The second representation. the one with which we will be concerned. is isomorphic to the graphs briefly described above. It consists of nxnxk matrices c with elmnents cu“ (id = l. 2. .... n; u= 1. 2. .... k). 11 where c1 .111 = O or 1. le impose the restriction that \gcidu = 0 or 1 for each ordered pair (1.1) of distinct individuals and by conven- tion we set cu“ a 0 for all i and u. If. for an ordered pair (id) of distinct individuals. one of the 1: distinct types of relations does not exist. then the null relation is mandatory and is represented by 61.111 = 0 for every n = l. 2. .... k. If one of the k relations. say the nth. exists between the ordered pair (1.3). then c110. '3 1, 613' ' O for V* ll. To establish notation. for each u = l. 2. ..., k. we let getfi for ..ch 1 = 1. 2’ e00. n! ”d 1.1.1 '3“ : éciJn for 0.911 J : 1. 2. OeO. n. and we form the n-dimensional marginal row and colunn total vectors and 5 (1'1“, 1'2“, .... tn“) 1.1.3 :11 : (31“, ea“. .... 'nu) respectively. where the superscript u is an indul. Also. we let n tn 5 girl“ : 5.3“ where the equality is an obvious consequence of the definitions. l'i'his should not lead to confusion for this notation will only be used in connect ion with the n-dinensional vectors 5 and .s_ . 13 In many sociometric tests each individual is asked to classify each of the others into one and only one of (k+l) categories. k>,l. with no restrictions on the nunber of individuals to be classified in a given category. Using the notation above. we point out that ”in is the total number of individuals classified by individual i into the nth category and that 'Ju is the amber of times that individual .1 was classified by other individuals into the nth category. In such sociometric tests it is of interest to how the exact probability distribution of the n-dimensional marginal total column vectors _s_“ (u I l. 2. .... 1:) given the n-dimensional mar- ginal row total vectors 5“ (u a l. 2. .... k). In order to deter- mine the exact probability distribution (under the assunption that all outcomes in the sample space are equally likely) a method of counting the possible outcomes must be developed. In the matrix representation this mounts to the problem of determining the total nunber of distinct nxnxk matrices c with elenents cu“ = O or 1 such that k 12195 Equuél for each ordered pair (1.)). i=FJ. 1’1 g le 2. '0'! n 0 1.1.5a fori=l.z. ....nandu=l.2. ....h. II 0 ciiu ' 1. 2. eeegk’ n 1.1.4 $1613“ ’- riu , 531.1“ 3 .311 for each u 13 where r111 and s3“ are respectively the ith and Jth components of the given fixed n-dimensional marginal row and colunn total vectors 5‘1 and 3“ . In order to determine this number it was necessary to form a product space which contained as a subspace all the nxnxk matrices satisfying the restrictions given by 1.1.3 and 1.1.4 . 1:2 The product ppm. Consider first a fixed but arbitrary relation. say the nth. for each fixed u (u = l. 2. .... k) there exists an nxn matrix which we denote by On. This matrix has elasents c1 Ju = 0 or 1 (1*J:1.1 = 1. 2. n) and onus o for 1 = 1. 2. n. Associated with each on are two fixed n-dimensional marginal row and column total vectors 5“ and g“ respectively. defined as in 1.1.4 . Thus. for fixed u. these matrices on exhibit binary rela- tions between ordered pairs of individuals . “1h a mathematical model was considered by Rats and Powell [5]. Using their methods we can find. for fixed but arbitrary g“ and 3“. “Lg“. 3“). the total number of distinct nxn matrices cu with elwnents on“ 1'- 0 or 1. subject to the restrictions 1.2.1 “11;; 0 101‘ ‘11 1 = 1. 2. ..., n and )5 f - 1e2e2 leciju = 1'1“ . 1:161:11 "' 8:“ s 14 where ”in and s1“ are respectively the ith and Jth components (of 3;“ and 3“. Thus. for each u = 1. 2. .... k. there exists a space of «(5“, 3“) matrices (.3u each subject to the restrictions 1.2.1 and 1.2.2. Ie now form the product space of the 1: spaces and obtain a space consisting of «(51, 3}) r652. pan-1Q}, 31‘) distinct elanents each of which is an nxnxk matrix C. we denote this prochict space by C = {a}. Many of the matrices in the product space will. in general. violate the restriction “if”... fl 1 for every ordered pair (1.1). Is stated previously that the purpose of forming this product space was to have a space which contained as a subspace all those nxnxk matrices c satisfying the restrictions given by 1.1.3 and 1.1.4. Is now show how to isolate this subspace. The product space C can be decomposed into mutually exclusive and exhaustive subspaces Co. Cl. 62. .... where Cu is that sub— space of 6 containing all those nxnxk matrices for which $101111 )1 for exactly m distinct ordered pairs (1.)). 14:3. It is easily seen that these subspaces Cm are mutually exclusive and they are obviously exhaustive for anv a. . Thus C = 601' q'i' 82+ . men at = 0 this implies 1.1.3 and since 1.2.2 is identical with 1.1.4 it follows that Co is the required subspace. ls denote the nunber of nxnxk matrices in CO by H(_r_1. _s_1. .... 5k. 3k). 15 Definition 1:2.1 Let 11(51. 31. 5k. 3k). 1:; 1. be the number of distinct nxnxk matrices c in the subspace GD of the product space 6 . In relation to the results of rats and Powell [5]. we point out that.for k: l. H(_i_' . 3) 5 1(5 . -s_). There is another nunber defined on the subspace co for which we shall have frequent use. This number. which is defined below. also involves a fixed but arbitrary set of I specified distinct off diagonal positions (iman). m = l. 2. .... I. Is denote this set by {1.4“}... Definition 1.2.2 11 1 (£1. 3}. _r_“. 31‘). 1:21. is the {... L number of distinct nxnxk matrices c which. in addition to belong- ing to the subspace 60 of the product space 6. have “:1 cu“ = 0 for each pair (1.3)6 {imdm‘ho Every theorem. lema. and corollary in this paper is dependent upon operators 3% . In the next section we will define these operators and will prove a most important learn with regard to them. 1.3 The operators “of. we require operators {if (u = l. 2. .... 1531.1: 1. 2. .... 11). These operators act independently on the n-dimensional vectors 5“ and _s_“ and are defined by 15 r ’ an .1 191(51' 1. .... 5W1. 3W1. rlu' .... r '1u’ iu' ru-b + rk .3 . see. .nu. - 1! 3“ 19 see. - O -) 1.3.1 3 1 11']. W1 _ (51. _s_ . .... .1; . g . r1“. .... rm- 1. .... rm, n+1 u+1 k .lus secs .3“. 1s sees .nui 3:. s a , ..., f. .3- )- That is. letting grin = tu. as before. the effect of the operator .9": on (£1, _s_1. .... g“. 3“. .... 5k. 31‘) is to replace ‘. the double partition r“. s“ of t.1 by the double partition of tn-l with the ith part of 5“ and the Jth part of g“ each reduced by unity. l'or any fixed it (u = 1. 2. .... 1:) these operators are pre- cisely the operators .51 defined and introduced by rats and Powell [5]. Hence. for fixed u. they have the same properties as the d", , namely. they are associative and comutative under addition and multi- plication. Since they act independently on their respective n-dimen- sional marginal total vectors 3;“ and _s_“ . they can be applied as operators on (51. £1. .... f. 31‘) in any order. Furthermore. the above properties serve to specify the effect of every sun of monomials In:pt+kuzpén1+mnnka3m1+---+-,..1+1... 1lift" ooo‘l'flk_ 1+1 of the form 1.3.2 :fia+eee+% k 1+...+%. .1101" mvbo.“1, 2’ see. k. 17 as an operator on (51. 31. .... 5k. 3k . Is further note that the amber of non-trivial terms is finite for finite partitions of t“ (u = 1. 2. .... 1:) since. for a function G. we have for every u Jl+h (a «£1.21. a ..o ‘5 {(3 “9"”(1-1. .1, 33:. 3}}50 1f h>0 and m 8 mink“. .311)“ 1 03.3 Equation 1.3.3 is written dually in order to exhibit the manner in which the ‘9: filter througl a function G. Finally. it is to be noted that any identity among Sukhatme's [3] 1(51. 3}). sets “d Powell's [5] «(£19 £1)s H(£ls 31s see. ‘2‘. 3k). 8nd 8&1: 3m} (£1. _s_1. .... 5k. 3k) is unaffected by the application of O I . these operators. since the operation on any of these functions is an operation on the partitions involved. to next establish inverses for certain operators. Lanna 1.3.1 The operator (1+ £2 fig). 15. hf: t. has an inverse. uzl left and right. given by (”33.94 (g...) (3;...) Proof: Using the associative and comnutative properties of the operators. it is easily seen that h J h ad)“ - h J)m+n (“=21 91).(‘E:1 u i " (3:3 191 . 18 To show that it is a left inverse. we have (1+; fl)-(i+n:°1 Applying the successive terms of the inverse, as given in the right 1.3.6 member of 1.3.4. first to the first term within braces end then to the second. we obtain. using 1.3.5 -1 It follows that €+uél 3‘11) . lé hé 1:. is a left inverse. A similar proof shows that it is also a right inverse and Loans 1.3.1 is proved. 1.4 The three stages of develepment. Up to the present time we have been developing a notation to be used for a general case of (k+1) categories. The reader will no doubt be surprised to find that in the next section we consider the particular case I: = 2. This is partly due to the method of development. The main purpose of this paper is to extend the count- ing theorem of fats and Powell [ 5] which corresponds to the case i: = 1. a first natural step in extending such a result would be to consider the case 1: = 2 (three categories). is a matter of fast we comenced by consideringia special case of k = 2. nanely where each individual is asked to classify exactly one individual in category one. exactly one other individual in category two. and the ruaining (n—3) individuals in category three. the null category. This case is given in the appendix and the reader w benefit by first examining this case. In most mathematical theories there is an underlying evolution process which is very evident here. In the appendix a counting theorem for the case described above is proved. This proof is constructed around an inherent feature of the special case. That 1'. since r11 = r12 = l for all i = 1. 2. .... n. this implies 19 1:- -n.....-II- "r'w- 5's .. . .. .. 20 that. for fixed 1. there is 23. 2’25. 9_n_e. ordered pair (1.1) such that c131 = c132 = 1. In the appendix. c131 = c132 = 1 is referred to as a coincidence at (i.J). In the next step. where the restrictions r11 = r12 = l are removed and 51 . 31 . f . 32 are more general n-dimehsionsl marginal total vectors. there is also an inherent feature. For any fixed ordered pair (1.3) of distinct individuals the only war in which 4 : the restriction. 3131+ ciJZ‘l’ can be violated is for can c133 = 1. However. this can happen more than once in each row and therefore this case is different from the one considered in the appendix. This is the case taken up in the next section. The proofs are quite comlicated. however. and for this reason we progress to the final result through a long series of Lamas and CorOllaries, hoping to preserve the continuity. Finally we come to the general case, k>l. in section 1.6. The proofs here are quite similar to those in section 1.5. However. again there is an inherent feature which distinguishes this case from the preceding cases. For a fixed ordered pair (1.3) and k) 2. k the restriction e1 3115‘1 can be violated in 2k-k-l ways. u-:l i.e.. the nunber of was of choosing two or more things from anong 1: different things. thus. for fixed (1.3). instead of having only one wsy of violating the restriction. we have mam ways and consequently the method of attack must be changed. 'e now consider the case k = 3 . 21 1:5 mg, 21, £3. 33) ss abilinear form of «(51“, 3}“) andrl(_1:_2“. sf) consider now the case where there are three types of relations between ordered pairs of distinct individuals. I: = 2. Let there be givm two sets of arbitrary but fixed n-dimensional marginal row and colunn total vectors :1. 31 and £3. 33. These vectors are sub3ect only to the restrictions imposed by the conditions that °iJl+ c1335: 1 for every ordered pair (1.3). 111:3. 1.3 = l. 2. .... n. Ior this case. if a type 1 relation exists between the ordered pair (1.3) of distinct individuals. then c131 = l; similarly for a type 2 relation. c133 3 1. If neither a type 1 nor a type 2 relation exists between the ordered pair (i.3) of distinct individuals then c111 = c132 = 0. vs wish to find HQ}. 3}. £3. 33). which by Definition 1:2.1 is the number of nxnx2 matrices c in Q. In general the product space 6 will contain nxnx2 matrices for which °131+ c1 .13 : 2 for one or more ordered pairs (1.3). Such matrices violate the restric- tion that c131+ c1.12 s. l for every ordered pair (id)- In order to find 3(51. 3}. £2. 32) we first consider a fixed but arbitrary pair (1.3). 1+3. 1.3 = 1. 2. n. in the nxnx2 matrices c of the subspace 60. Associated with this specified ordered pair (1.3) are two numbers cu1 and c“2 . i.e.. k = 2. and since we are concerned with the subspace Co. the ordered number pair (0131. c153) can take on one and only one of the three values (0. O). (0.1) and (l. 0). Thus. with respect to the ordered pair (id) of distinct individuals. the subspace Co of the product space C can be divided into three mutually exclusive and exhaustive classes. This decomposition is given by the following lama. Lama 1:5.1 Iith respect to the arbitrary but fixed ordered pair (1.3) of distinct individuals. the subspace Co of the product space C can be decomposed into three mutually exclusive and exhaustive classes represented by 1.5.1 my}. 31. :2. s2) : (1+1di-i- 2o(f)3{1.31(£1.31.52.33)- Proof: Expanding the right member of 1.5.1. the first term is. by Definition 132.2. the nmnber of nxnx2 matrices C in Co for which c131 = 01.12 = 0. The second term is the number of nxnx2 matrices C in Q for which c,“11 = l and c1J2 = 0. since if a type one relation is fixed between the ordered pair (1.3) of distinct individ- uals then r11 and '31 must each be reduced by unity. This is precisely the effect of the operator Pg on (51. _s_1. £3. 33). Similarly. the third term is the number of nxnx2 matrices c in the subspace CD for which c1” "-' 0 and c132 = l. Iith respect to the ordered pair (1.3) this exhausts the possible pairs of values for the amber pair (c131. cu?) in the subspace CO and hence the Lemma follows. --1 Applying the inverse. (1+ 10¢ ‘1' gig) . given by Leuma 1.3.1 with h 2 2. to both members of 1.5.1. we have Corollary l:5.l.l For an arbitrary but fixed ordered pair (1.3) of distinct individuals. the nunber of nxnx2 matrices C in the sub- space 60 of the product space 6 for which c131 = c1""; = 0 is given by -1 (1+ 92+ re) ......ig. 1.5.2 33.33551. 31. 3;". .33) The following corollary could be omitted completely. It is included to illustrate for single ordered pairs (1:3) the procedure to be followed later on for sets of ordered pairs (1.3). i‘o be more explicit. rheoranl.5.3 (near the end of this section) emerates the nunber of nxnx2 matrices c in each of the subspaces Co. Cl. 6;. ’°° . The following corollary gives explicitly the number in CD it is a special case of Corollary 1.5.2.2 which gives the nunber in em for m=1. 2. . The procedure is to fix a type 1 and a type 2 relation between the arbitrary but fixed ordered pair (1.3) of distinct individuals. If this is done for every possible ordered pair (1.3). 11:3. and the sun is taken over all possible pairs. the result is Corollary 1:5.1 .2 The nunber of nxnx2 matrices c in the subsP8¢° CI of the product space 6 is given by J 1.5.3 5 1&3 ?(112°§(1+1°‘£ +594 30:1. .21- £2. 33) - Proof: By Definition 1:2.2 Hi1.1}1(_r_1. 31. £3. :2) is the number of nxnx2 matrices c inthe subspace CD for which c1.11 = c132 = 0. Is can assme without loss of generality that ru. s31. 1‘12. and s32 are greater thui or equal to one. Is now fix a type 1 relation between the ordered pair (1.3). thus making c131 1' 1. Since rm and en are respectively the amber of type one rela- tions in row i and column 3. this one must come from these com- ponents of 51 and 31. Similary. if we fix c132 = 1. then r12 and s3; mist each be reduced by unity. This removal is effected by applying the operator 10911 N to (£1. 31. £2, 32). It follows that 1°"? m ngdhLL-l. _s_1. £2. 32) is the nmnber of nxnx2 matrices c in the product space C for which c131 Z cuz I 1. and by defi- nition each of these matrices belongs to 61 . Applying corollary 1.5.1.1 to the left member of 1.5.3 we obtain the riglt member for each ordered pair (1.3) and if the sum is taken over all possible locations we exhaust the cases for which c111 ='- c132 = l. The corollary is an immediate consequence. Is now extend Lanna 1.5.1 to in specific distinct off-diagonal positions (imam). m = 1. 2. s. 1.4”“ . again using the notation {imhk for this set. Lanna 1:5.2 Iith respect to the fixed but arbitrary set {5'3”}11 the subspace CO of the product space 8 can be decomposed into 25 3“ mutually exclusive and exhaustive classes enunerated by {fiéw‘cw‘ifl} 1 2 2 Hés-JSSJEI' i .214.)- Preof: In the discussien which follows it should be understood that some of the classes may be vacucus. It is obvious that there are 3' distinct terms in the right member of 1.5.4. There is an isomorphic: between these terms and the totality of number pairs (c111. c132) for (1.3)e £5.33“. The subspace under consideration is 60 and therefore for each (1.3) £- {imsdm3l we have “131' c132) = (0.0). (1.0) or (0.1). Thus. for each of the I specified positions (1.3)6 {imdfir the amber pair (0131. c132) can assune am one of the three distinct values and it follows that there are 311 nunber pairs on the specified set. Consider the following general set of a nunber pairs on the specified set of positions finding“. nanely, cipJpl : l. cipdpz : 0. :O.C112:1, for p=m1+l. for p=19 2. ...'m1; 01 Jpl pp P x = = 1. n1+2s sss, m1+% ‘,' “d cipapl cipjpa 09 for p “1+5..- m1+m2+2. .... Iii-mafia . where m1+nz+m3 3 11- since 26 n (r1. s1. :3. 32) enmerates a set which. by Definition 1.2.2, {5'1“}11 ' ' "' has c131 = c1.12 = 0 for (1.3)5 £1“.3.}u. we can proceed as follows. Fix a type one relation at the .1 (0 5 m1 5.!) positions indicated above. then r1 1 and s‘1 1 must each be reduced by unity for p 3 P P 1. 2. .... m1 ; these reductions are effected by applying the opera- tors P‘i: mm to £1. :1. Similarly. fix type two relations at the m3 (0 .4. m2 fill-m1) positions indicated above. then r 2 and these reductions are effected by applying the operators guilt: .. . l gar?!» to 52“ 2. Carrying out these reductions simultaneously. Inrims we would have 1.5.5 (13111-"91211‘1‘1'1m204mlimznl-52. £2)- “.{tn tho-'1’ This is readily seen to be a unique term in the expansion of 1.5.4 for it is obtained from the I distinct factors by taking the speci— fied I“? from :1 of the I factors. taking the specified KP P from me of the u factors (distinct from the preceding m1). and taking 1 from the winning 11 - in1 - m2 factors. Since this was done in complete generality and since the converse is easily demcnstrated. it follows that there is a one~to-one correspondence and the lemma is proved. 27 Repeated application of Lanna 1.3.1 to both members of 1.5.4 gives us. without further proof Corollary 1:5.2.l The number of distinct nxnx2 matrices C in the subspace 60 of the product space e for which c1 .11 = c152 = 0. where (1.3)5 {111.44% is given by - l .1. '1 1.5.6 Hi1m'1n}u(£1. £11 52' «32) ' IU;(1+E1M+ £112) 1 Is now proceed to Corollary 1.5.2.2 which is a generalisation 01' Corollary 1.5.1.2. That is. between each ordered pair (1.3). (1.96 {54.3“ we fix both s type one and a typo two relation. Corollary l:5.2.2 The umber of distinct nxnx2 matrices- C in the product space 8 which have cut: e132 = l. fer every ordered pair (1.J)€ [ind-3“. and so coincidences elsewhere. is given by 1.5.7 {13; 981: gtgngmhguul. 31, £2. 33) : Jm m a- - ...-.- 9.“ MEX” 985+5’81.) “51-31-3122). Proof : In the diseussiea which fellews it should be understood that some of the classes may be vaeuous. Bynefinition 1.2.2 B (r1 s1. r3. s3 is tho {E’Jmh‘ ' " "' ") number of nxnx2 matrices C in the subspace 80 for which 28 c131 = c132 = 0 for every (1.3)6 £5.34“. therefore we can proceed as follows. rix a type one and a type two relation between each ordered pair (1.3)6 {5411314 and it follows that c131 = c132 = l for each of the ordered pairs. If this is done. then the corresponding components of the n-dimensional marginal row and column total vectors (£1. 31. :2. 32) must each be reduced by unity. but this is precisely the effect of +1- 931: '2‘: 01 (£1. 31. 12. 33). This proves the corollary with m=l regard to the left member of 1.5.7 and the right member follows directly from Corollary 1.5.2.1. The preceding corollary applies only to the specified ii. .33“. 1121. e“ is that subspace of C which contains all the matrices having c1.11 + c112 ) l for exactly I positions (1.3). i#3. and no coincidences elsewhere. Thus matrices enumerated by 1.5.7 belong only to C“. It follows that if 1.5.7 is smed over all possible locations of the sets {ho-1.3g. the result is the total number of matrices in the subspace C“. for fixed l the amber of distinct sets {5.3“}! isthe number of ways in which 11 distinct off-diagonal pesitions can be chosen from n(n-l) Positions. nasely (Mi-1)). Is denote the sum over these (“hi“) sets by Using this notation and the previeus remarks {1mm we may write Corollary 1.5.2.3 The amber of distinct nxnx2 matriees C in the subspace 3. (112.1) of the prochict space 8 is given by ,— va~ _ ’ r: rkn‘ a... ‘5‘..fl-’ *9. ‘f—TF-r.‘ nu. Z{(. “a- “3) {1- so.” 3} {in Ja}. {5.3. mg; m.» .22. rate. £2. Proof: Each of the (“flamm- in 1.5.8 is. for fixed u. of the 1.5.8 form given in Corollary 1.5.2.2 and hence gives the nmber of tuna matrices C in the product space 6 for which c131 = c132 ‘2 l for each ordered pair (1.3)6 £5.43". Since each such matrix violates the restriction that °1J1+ c132 5 1 for exactly I! ordered pairs (1.3) . each belongs to C“ and since we are summing over all possible locations of the sets {5.3. we exhaust the possibilities for the subspace en. The corollary is an immediate consequencewith regard to the left member and applying Corollary 1.5.2.1 to the left member, the right member follows and this completes the proof. It was shown in section 1.2 that the total number of distinct nxnfl matrices C in the product space 6 is «(51. gl)rl(£2. £2). i.e.. k 3 2. where the It function is the function introduced by late and Powell [5]. At the some time we showed that the product space could be decomposed into mutually exclusive and erhmative subspaces. that is C = Bod-31+ g + . The following theoran gives a complete enmeration of the decomposition of the product space 6 into subspaces . Theorem 1:5.3 If 51. _s_1 are any two n-part. non-negative. ordered partitions of t1 and 52. _s_2 are any two n-part. non-negative. ordered partitions of t2. then an exhaustive enumeration of the nxnx2 matrices C in each of the subspaces 8“ (1| )0) of the product space 8 is 617011 by J J 1 ,2 : 7T 1°‘1 f‘i 1.5.9 11(51- _'. )rvl‘zv .) 1*Jé+ WE“) x HQ}. 31. 3:2. :2). Proof: Inpanding the product in the right member of 1.5.9. we have J 1.5.10 8(51. .1. r3. f)+ Z =4“: £1 ~ mrl. 2.1. 1'2. .3) +2 1%“ 10“}: ‘ 36.21-13.12 gran}: n=1 1+ 1“i.+ fill the first term is by Definition 1.2.1 the number of distinct nxnxz matrices C in the subspace 80 of the product space 8 . It can be seen from 1.3.4 of Lama 1.8.1 that the inverse (1“!- lui+ girl is the algebraic inverse and hence that the second term of 1.5.10 is the number of distinct nxnxz matrices o in the subspace 81 by Corollary 1.5.1.2. The third and succeeding terms 3'1 are precisely those of the right munber of 1.5.8 of Corollary 1.5.2.3 for I = 2. 3, . respectively. It follows that the third and succeeding toms enmerate the number of distinct nxnxz matrices C in the respective subspaces 82, 83' etc. The theorem is an imnediate consequence . The main problun under consideration is to find the nunber of distinct nmxB matrices C in the subspace 30. which by Defini- tion 1.2.1 is 3(51. _s_1, £2. 32). For each factor in the right masher of 1.5.9. it is easily seen that J #3 20a _ 1+ 91+ 24 If this substitution is made in 1.5.9 and Lemma 1.3.1 is applied to 1+ both sides of 1.5.9 for each factor in the nmerater and denominator after substitution, we obtain Theorem 1:5.4 If 51. 31 are any two n-part. non-negative. ordered partitions of t1 and £3. 33 are eav two n—part. non-negative. ordered partitions of t3. then the number of distinct nznxz matrices C in the subspace Go of the product space 8 is given by 1.5.11 3(51. 3}. 53. 32) 3 TT 1=\=J 1+p< +pt 2 where the operators lag and 8‘: operate on T51. 11) and [\(gz. 2, ). respectively. Since there are no tables published which give the value of the ‘01. I1) functions. the preceding theorem would hardly lend itself ps- ~_=‘..L A. " V- ... — .— .L.___.__—- #mA. ’ I. ‘, _', '. 17 32 to computation. However. Rats and Powell [5] proved that 1g, 3) n 1 -1 : 130411) “£1 3), where thO'll-B] 1(g. 9 function is tabu- lated in reforeneo [8] for partitions of t up to t = 8 and also in David and Kendall [1] for partitions of t up to t = 12. In section 1.3 we stated that the operators nod were precisely the 1 same as the 61 introduced by Rats and Powell [5]. Therefore. for u-'-1.2.wehavo n '1 i 1.5.12 «(35“. 9.“) = H 1+ .9") “a“. 2“) ~ i=1 Making this substitution in Theorem 1:5.4. we obtain the following corollary without further proof. It should be noted that in 1.5.13. below, the product in the denominator is now over all i and .1. Corollary 1.5.4.1 If 51. 31 are shy two n-part, non-negative. ordered partitions of t1 and 53. 33 are any two n—part,.non-negative. ordered partitions of t2. then the number of distinct :1an matrices C in the subspace 60 of the product space 8 is given by 1+ ll 1.5.13 3(51- £1. £3. £2) 3 1 (“Hal 2‘13 MEI. 31)“!22’ 2.2). 1.: 1+ 1‘91” {1] where the operators 1“: and :94 operate on “51. 31) find “£3. 2.2). respectively . to now consider a special case of Theorem 1.5.4. This case is identical with the one considered in the appendix. as can be seen by comparing the following corollary with TheoreaLSA in the appendix. The following corollary is applicable to sociometric tests involving l'._ __a77 three categories (1: = 2) where it is mandatory that each individual classify exactly one individual in category one. exactly one other individual in category two. and the ranaining (n-a) individuals in the third category. Per such tests. we have. in the matrix model. 31 = 53 = (1. 1. .... 1). Is shall denote this n—dinonsionel vector of n ones by 1“. Ivory ‘9‘: involves both a row and column index. therefore it follows that any term which involves (94)“. where 1112-2. is trivial. since rm 3 1 for all 1. Therefore. it follows that 1+pif+éxf _ J 55561-53) ‘ “ “i + ”‘ where all the terms in R are trivial since they involve powers of 1.5.14 9‘: greater than one. Equation 1.5.15 of Corollary 1.4.5.2 below is written dually. First Bun. _s_1. In. 32) is expressed in terms of [1(51, :1), 1 Z 1. 2. by substituting from 1.5.14 in 1.5.11. In the second form. obtained by the additional substitution from 1.5.12. an“, “31, 1“, 33) 1. .3. pressed as a function of 1(51. :1). i = l. 2 . to now give the corollary without further proof. coronary 1.5.4.2 1t r11 = r13 = 1 (1 = 1. a. .... n). and 31 and s2 are any two n-part. non—negative. ordered partitions of n. then the mnber of distinct nxnx2 matrices C in the subspace 80 of the product space 8 is given by 84 1.5.151n._1.1n. s3) " {1m 1 - pi§§)}”rk(1.s1)V(1“.gz). :£;I( - 1 1M¥1)19r1=1 1’N-£1+p1 fi)}x A( 111 ’31)“ 111 032) ° 35 ska) . 1.6 11(51. :1. 5“. 3“) as afunction of «51¢, 31“) p651! , In this section Theorem 1.5.4 is generalized. The generalisation is from three types of relations between ordered pairs of individuals. in a group of n individuals. to (k-t-l). k>1. types of relations between ordered pairs of individuals. It will be of interest to compare the method used in this section to find 3(51. 3}. .... 5k. 3k). k>l. with the method used in the case I: = 2 of section 1.5. In both sections 1.5 and 1.6 there is the necessary restriction that “g cu“ = 0 or 1 for every ordered pair (1.3). 1* J. In section 1.5 there was associated with every ordered pair (1.3) of distinct individuals a number pair (c131. one). It is easily seen that. for any (1.3). the only nunber pair which violates the restriction “goth : 0 or 1, l: = 2. is the amber pair (1.1). However. when k >2. there is associated with every ordered pair (i.J) of distinct individ— uals a k-dimensional vector (c131 ’ciJZ' .... c1 51‘). It is obvious that the necessary restriction valet-1“ 3 c or 1 is violated whenever two or more °iJu 2 l. u = 1. 2. .... k. It follows that,whereas when k = 2 there was only one possible violation. nanoly (1.1). there is now a total of 21‘ - k - 1 different ways in which the necessary restriction could be violated for every pair (1.5). aka. rm. implies that a method different from that used in section 1.5 will have to be used in this section to eliminate from the product space 3 all those matrices violating the necessary restriction. Using the definitions and notations previously established we now proceed to Laura 1.6.1 which is a generalisation of Lemma 1.5.2. Leann 1.6.1 Iith respect to the arbitrary but fixed subset £5,545“ of I off diagonal positions. I21. the subspace Go of the product space 8 can be docomptpsed into l+ 1 mutually exclusive and ex- haustive classes enumerated by 1.5.1 3(51. 31. 5“. 3“) ={fi'lé+ $1 “0:31:31: azimd-‘gu‘glt 210 "N 3:" :1) Proof: Since there are (l:+ 1) distinct terms in each of the n a)..- tinct factors in the rig1t member of 1.6.1. it is obvious that there are (l + k)“ distinct terms in the expansion. The method of proof '111 be to and. the (1+ 1:)“ terms into n+1 subsets and prove that each of these n+1 subsets corresponds uniquely to a class of nnxk matrices in the‘subspace Co. The n+1 classes will be shown to be mutually exclusive and exhaustive with regard to the arbitrary but fixed set {13.3.13 of II distinct off diagonal posi- tions. By definitions 1.2.1 and 1.2.2. mg‘. 31. 5". f) and ni‘m'JmSlul’ £1, ..., 5k, 3:) are defined on the subspace Go of the product space 6 . Therefore. for every ordered pair (1.3) of distinct individuals. g cu“ = O or 1. lith this established it is obvious that there are 1: ways in which 5 ciJu 3- l and only 87 one wu in which g cm“ 2 0 for each ordered pair (1.5). 1*.1- lo shall carry out the decomposition of CD into n+1 mutually exclusive and exhaustive classes and simultaneously show the unique correspondence of these classes to terms in 1.6.1. The sore class. which we denote by Cm. is that class containing all those nxnxk matrices for which 5 ciJu 3 O for each of the ordered pairs (i.J)£- {13.3.}... It is not difficult to see that this class is represented in the expansion by taking 1 from each of the I factors in 1.6.1. thus obtaining 31‘0"; (:1. £1, .... 5k. it). The first class. 601’ contains all those nrnxk matrices for which .E c1111 1’ 1 for 22°31 one of the ordered pairs (i.1)e {in-Jala- n is quite evident that this can be done in (‘1'): ways since the single ordered pair can be chosen from H pairs in ('1‘) ways and the type of relation for this pair in an of 1: ways. This mounts to taking one from all but one of the If factors in 1.6.1 and from that factor choosing aw one of the k different .94. Proceeding in this manner. we now consider the 3th class. con. 0 s 35!. This class. 603' contains all those nxnxk matrices for which 5 °1JII 1' 1 for exactly 3 of the ordered pairs (1.3)6 £5.33“. This can be done in (391:3 me. since we can choose 3 ordered pairs from I ordered pairs in (g) ways and once chosen we can choose the type of relation for each in 1: ways for a total of (aka ways. This mounts to taking a one from u - a of the I factors in 1.6.1 and exactly one ‘9‘: (which con be chosen in any of k distinct ways) from each of the remaining 3 factors. These steps are reversible. hence the corres- pondence between classes 803' O 2 B él. end terms in 1.6.1 is a one-to-one correspondence. There are obviously I + 1 terms of the form (at! in (1 +k)“ and from the manner in which 803 is defined. the classes are mutually exclusive and exhaustive. The launa is an mediate consequence. Repeated application of Leanna 1.8.1 to both members of 1.6.1 gives us. without further proof Corollary 1.6.1.1 The nunbor of distinct nxnxk matrices C in the subspace 80 of the product space 6 for which 11% c1,“ = 0 for each ordered pair (i.J)& {5.3,}3. I ?_1. is given by 1.6.2 nftm.3.h(51' _._1. .... 2:“. 3:) s 1:];(1-1- $3 3‘13) From this point on the proof given in this section differs from 1 3(3)}. :1- °°°s its ...,k) ’ that given in section 1.5. Io now wish to ornine the wave in which $ 613“) 1. In the case k = 2. of section 1.5. this could happen in only one wq. namely ciJl = one = 1. Iith k >2 the nusbor of ways is 2: - h - 1. for each ordered pair (1.1). In this connection we new state the following definition. Definition 1.6.1 To shall say we have a w—tuple coincidence at (1.3).25wék. if gen“: w. It was shown in section 1.2 that the product space i 6 could be decomposed into mutually exclusive and exhaustive subspaces G“. I 20. so that 6: 80+ 61+ 62‘1"" , at the one time 8“ was defined to be that subspace of the product space 8 which contained all those nxnxk matrices C for which g °i3u>1 for an_c_t_11 I ordered pairs (id) of distinct individuals. By Definition 1.2.2 Bgmd-‘gugl. £1. .... 5k. 31) enumerates a subspace for which c1 3“ = 0 for each ordered p313 (1.3) G u: {fidér Is now proceed to fix a w-tuple coincidence (2 £- w f. k) at one of the u positions given by the ordered pairs (i.J)e{i..J-75l. Let the w-tuplo coincidence at (id) consist of the relations 11*. v = 1. z. .... w. 2 2.5.1:. then at”, = 1 for v = 1. 2. .... w. If this w-tuple is fixed at (1.3). then rm' and 'Ju,' v = l. 2. .... I. must each be reduced by unity. This can only be done by the Operator “lag “Bag ...ng‘f applied to (51. 9}. g“. 9}). It fol lows that ' 1.6.3 Eugigaih’fiI‘é' £1, .... g, 3k) space C which contain : 1 is the amber of nxnxk matrices in the product the particular w-tuple coincidence e1 3111 = o1 .1112 = - cu“. 5‘ position (1.3). we point out that. for any fixed w,(2 go 5.1:). there would actually be (5) distinct wave of choosing a w-tuplo coin- Cidenco. Therefore. to include all possibilities for the position “a”: it is necessary to sun 1.6.3 over u1 1- I- point Out that 1!! TT “'31) there are (1 + 1),: = 3]: distinct terms and in 1 + “in 91 there are (ll-P1) distinct terns. [O are merely choosing those resulting in w-tuple (3.... wék) coincidences at (1.1). corollary 1.6.1.3 applies to each position (1.1)6- 8.4.“. 11?. l. to new wish to enumerate the nxnxk matrices having a coincidence of multiplicity 2. 2 at every ordered pair (1.3)5 {Why me can be accomplished by taking the Product of factors of the form given in 1.6.? over all ordered pairs (1.3)6 firing“. Since it is desirable to express this'as a function of mg. 31. “.... f. f). we employ 1.6.2 of Corollary 1.6.1.1 for the right member of 1.6.8 below. to point out that the total nunber 01' pessibilitios of having some type of a w-tuple (29.51:) coincidence u «or: (tune [1.6.3.1 1- (2“ - t - 1)“- cerellary 1.6.1.4 The total number of unit matrices which have a coin- cidence of multiplicty _>. 2 at every (1.3)6 {5.533. ll :1. and no coincidences elsewhere. is 317011 b! I 1.6.8 Tr fi(:+ 9") - Jmu(1 + £13.11. {ime‘SlLElohl-u-v- 9 IF]. 43 The above corollary applies to the arbitrary but specified set of it off diagonal positions {"m'Jmh' I 21. for any fixed I this set can be chosen in (“If”) ways and we now wish to sun over all these possible sets {imam for an fixed 11?.1. to shall denote this on. for fixed 11. by . Therefore. earning the right 1114:1311 masher of 1.6.8 in the manner Just indicated. the following lemma is an imediato consequence of Corollary 1.6.1.4 and the definition of G“. note that we are now dispensing with the auxiliary function HttmoJJuLEI' 31' 'A'” g' ...k)' Learns 1.6.2 ror any M Z 1. the total number of nxnxk matrices in the subspace all of the product space 3 is given by 1.6.9 {1.5“} 3.1“;111(l+))°d:)'( 1 + Z :fi-nfl‘ftfi“ H(-1:1. £1 e 00-e_e_ -1 to previously showed that the product space could be decomposed into mutually exclusive and exhaustive subspaces. i.e. C 3 80+ 81+ 92* . The preceding lama gives the nmber in any of the Mspaces Brill. Therefore. if 1.6.9 1' “m“ on I, we would obtain a complete enumeration of the product space by subspaces el' ll 2. 1. with the exception of ea. The nunbsr in 80 is. by Definition 1.2.1. 3(51. 5.}. 1;“. 31‘). Hence. if the identity I operator is added to the m on 11 of 1.5.9, then the result would be a complete enmeration of the totality of the ((51. 31)”- “ff. sk) nxnxk matrices C in the product space 8 by subspaces 11’ I Z. 0. Consider the following expansion 1. 6.10 1T=1=1f"' 11-1 “191) 6*]: £1 19‘)“ ”$.29 1+ 263 1181 1 + - 91 1: +2 fi; jl+9€)-(1+1§1€d l$1.1. '11:, 1:) {1114132 1118 1+ 2 5i -. 11:1 .1. 1: 1: + Z: 11 1110+ \P‘i)’(1+§1 3‘1 1.1 1 11:31:) _ 1: .1 H(-..2- ...-.. 8‘4”“ "’1 1+ 59‘. The first term in right member of 1.6.10 is the amber of matrices in G . The second and succeeding terms are. by 1.6.9 of Lemma 1.6.2. ws that the number of matrices in e1. 63. .... Cl" It follo t: ‘1 44 the left member of 1.6.10 completely enmerates. by subspaces e . I Z 0. the totality of the «(51. .51)"- «(35. 3k) nxaxk matrices in the product space 6. The left member of 1.6.10 can be simplified since. for each factor 1:1: ,1. we have‘ 1 k 1 he ) 1.6.11 {1+ ug+fg'(1+§1gi : g+fi k 0 1 + g «4 1+ .5 s: ._ . 1 If 1.6.11 is substituted in 1.6.10 we have. without further proof Theorem 1.6.3 Given 1: pairs (5‘. 3‘). u = 1. 2. k. where g“ and _s_“ are any n-part. non-negative. ordered partitions of t“. t.‘ Z 1. then the totality of “£1. 3})"- r‘f. 3k) nxnxk matrices c in the product space 6 defined by these partitions is completely onunerat ed by k 11- .1 1.6.12 «£1. £1)°'° mg. 3}) 3 ll EM 3(51931o0'09g02k). i=1: Hf?“ u=1 We are primarily concerned with the umber of distinct nxnxk matrices C in the subspace Co of the product space 8 . namely 3(51. 31. .... 5k. 3"). It was ohown in Lemma 1.3.1 that. for 12115:. ”1° 13"". 01' (1+ “£1 ff) is the algebraic inverse. Each factor 11 right member of 1.6.12 is of the form given in Lemma 1.3.1. therefore repeated application of the lema to both sides of 1.6.12 gives us. without further proof. the following theorem. Theorem 1.6.4 Given 1: pairs (5“. g“). u = 1. 2. .... 1:. where g“ and 3“ are any n-part. non-negative. ordered partitions of tn. t‘1 Z 1. then the number of distinct nxnxl: matrices C in the sub- space 60 of the product space a is given by . + 6 1.6.13 HQ}. 31. 5“. 3“) =]T 1 “'1 u 1:] 1169.11)"- «(f-f). where M operates on (5“. g“). u = 1. 2. .... k. In the fans given by 1.6.18 we note that 3(31. 9}. . ... 5k. 3k) is expressed as a function of the 1m“. 34) introduced by Kate and Powell [5]. At the present time no published tables are available for their function T5“. 3“). Therefore. as in Section 1.5. we employ their counting theorem. nanely n - i -1 1g“. 9.“) - E(1+ 9:.) hr“. 311). for each u = l. 2. .... 1:. If this substitution is made for each Ti“. 3“) given in 1.6.13. then Theorem 1.6.4 can be written in the form given in the following corollary. Corollary 1.6.4.1 Given 1: pairs (3“. 3“). u = 1. 2. .... k. where a“ and 3“ are any n-pert. non-negative. ordered partitions of t“. tn 2 1. then the number of distinct nxnxk matrices C in the subspace cm of the product space 8 is given by i .) 1.6.14 3(51. 3}. 3;“, 5,") = J‘ELH “‘1 at 1071,31)... (if). U64- sf) where a“: operates on (5“. _s_“) u 2 1. 2. ..., 1:. IO 'call the readers attention to the product in the denominator of 1.6.14 which is over all pairs (id). including (1.1). Tables of M; . _s_ ) are available for partitions of t up to t = 8 in Sukhatme ['6]. Furthermore. Snkhatme [6] gives an algorism .for finding any 1(5 . a ). David and Kendall [1] give tables of M; . g ) for partitions of t up to t = 12. It can now be seen that Corollary 1.5.4.1 in section 1.5 is a special case of Corollary 1.6.4.1. nsnely the case k = 2. also. Theorem 1.3.4 in the appendix is a special case of Corollary 1.6.4.1. nunely k = 2 and 51 = 53 = (l. 1. .... 1). In this connection we point out that unpublished tables have been completed for 1(1‘. _s_) for values of n through 16 under the direction of Rats. These tables give the value of 110‘. _s_) for any 3. _s_ being n-part. non- negative. ordered partitions of n. It is for this reason that we next consider a special form of Theorem 1.6.4. The following corol- lary can be used in sociometric tests in which it is mandatory that each person classify exactly one other person in each of the k cate- gories. the remaining n-k-l being automatically classified in the (ht-1)st category. yer wuch tests 5“ = (l. 1. .... 1). for u = 1. 2. .... h. It follows that in the expansion of 1.6.13. any terms which contain a factor of the form (91)“: where m > 1. are trivial. Io now consider the expansion of 1.6.13 for this special case. 1+ 2519!: to: fixed 1 and j u- can be expressed as “E (1 + 3.3 j 4? elementary symmetric functions of the w by letting J J ‘1‘ = Zului ‘3‘ ...‘gg “1*‘3*...¢ur . Using this notation we have 1+ “a: -_- 1+“1 . 33(14- gf) 1+‘1+‘2+"' +5: Carrying out the division in a formal manner. we have 1- 1.6.16 l+a1+~~+et I1+a1 1+e1+a2+as+---+&r+~-+°n' 42 - ‘3 - "r" *‘1: Next we must subtract from the first remainder -a2(1+ a1+ 1.6.15 + “1:.2) . In this expansion we wish to retain only elanentary syn-’- metrie functions. rrom Sukhatme [8] we find that in the expansion 0f ame'. in tonne of monomial symetrio functions. that the only tom involving an elanentary symnetric function is C(lu'") = 11“,, and its coefficient is (01w). It follows that the coefficient of a, in the second remainder is ~1+ (5). furthermore the next term in the quotient 1. 233, i.e. —1+(g)= 2. If we accunlate the coefficient of 'r' for the third remainder. this coefficient will be ~1+ (5) - 2(g),for the fourth remainder —1+(£)- 2(3‘1-36). 1 etc. Proceeding in this manner we find that the coefficient of ‘r in the quotient is (-)r'1(r-1). and therefore we have from 1.6.15 .1 1+ {:1 9‘1 17(1 +94) where the remainder 3 involves only trivial terms. to shall write 1.6.17 1“" -.- 1 - e2+2n3-3n4+---(-) (bust-+1. the right member of 1.6.1? in the form 1+“: + B in the interest 48 of brevity of notation. Using this notation for each of the factors in 1.6.13. the special case of Theorn 1.6.4 can be written as c."11m 10604o2 1: 1'1“ = 1. for 1 = 1. 2. eoe. ‘ and“ = 1. 2. coo. k. and if 3‘. u = 1. 2. .... k. are an n-port. non-negative. ordered partitions of n. then the number of nxnxk matrices C in the sub- space 90 of the product space 6 is given by 1.6.18 11(1‘. 3}. 1‘. 3") = “'(1 +ai’r((1“41)"-V[(1‘-gk). where 1+“: is the non-trivial part of 1.6.17. Io I”int out that this corollary could be used without reference to tables 1: Theorem 1.2. page 29. in the dissertation of Powell [7]. is employed. His theoran expresses {(l‘. g) as a function of cynmotric functions of the non-sore s1 in 3 . In the next section we shall consider applications of the results of thi s section . PART I I APPLICATI 0N8 2.1 Preliminaries The main result of section 1.6 gives a method of finding the number. HQ}. :1. .... f. _s_“). of nxnxk matrices C defined by 1.1.3 and 1.1.4. Thus. for any h pairs of fixed n-dimonsional marginal row and column total vectors (5‘1. g“) u = 1. 2. .... k. we have a space co consisting of H(_1:1. _s_l. .... 5“. _s_") matrices C. The definition of these matrices was formulated by considering the possible outcomes of a class of sociometric tests. This class consists of tests in which each member of the group must classify each of the others into one and only one of (15+ 1) distinct cate- gories. k >,1. the categories being dependent on the criterion of the test and the number classified in each cateyry possibly restricted by the experimenter. A large number of sociometric tests are so characterised. lo now wish to consider some of the statistical aspects of the preceding results. We will define random variables within the realm of these tests and swine a particular hypothetical random variable. 2.2 Random variables defined on the matrices C. Usually. the number of individuals (n) and the amber of cats-- gories (bi-1) are takai to be finite. It then follows that the space of all possible outcomes of any given sociometric test can be so represented by a finite amber of nxhxl: matrices C. For an given test. let us call this space of all possible outcomes the sample space. He denote the total number of outcomes by I. ror this sanple space we shall adopt the usual convention and consider all possible outcomes as equally likely. Using this conven- tion we assign probability of l/ll to each of the l outcomes. resulting in a uniform probability measure on the sanple space. In some relatively rare cirsmstances we may wish to assign uniform probability measure on subsets of such a sanplc space; we here avoid this complication. Given this mp1s space of I nnxk matrices C we can now define various random variables over these 1! matrices. A random variable is a single valued. measure preserving mapping from its domain to its range space. The domain in this case is the sanplc space. The range space for the random variables we are considering will be appropriate subsets of the real line. Thus. for a given random variable 1. each of the II matrices in the sanple space has an image on the real line. this image being the value of the random variable x for the particular matrix. In general this mapping is many to one. that is. there are many matrices 0 resulting in the cane value of the random variable x. This map- ping induces a probability measure on the random variable since. for a given I = x. the probability measure of x is the probability measure of the inverse image of x in the domain space. It follows that if the dolnaia has uniform probability measure on it. then. using 51 the counting procedures developed in section 1.6. we can find the probability distribution of the random variable x. for any given sociometric test. the instructions with regard to making responses to the other members of the group will determine the sample space on which random variables will be defined; in other words. will delimit the universe of discourse. Is will consider a few of the sc sample spaces . 2.3 Some sample spaces defined by test instructions. In the discussion which follows. we assume that the restrictions on the nxnxk matrices C given by 1.1.3 and 1.1.4 are satisfied. Suppose the instructions to a sociometric test state that each person must classify d of the others in a first group. d more in a second group. etc.. for a total of 1: categories. lad $.n-l. and the remaining n-kd-l manbers are classified in the (k+l)st eate- Gory. For fixed d and k the corresponding natural sanple space consists of all nxnxk matrices C having 5“ = (d. d. .... d) for u = l. 2. .... k. For each of the sets of l: u-dimensional vectors 3“. u = 1. 2. .... k. we can find H(d‘. _s_l. d‘. 3.2. .... d‘. _sk) and thus the number in a subspace Co of the ample space. There are as many such subspaces co in the ample space as there are distinct sets of k n-dinensional vectors 3“. u = l. 2. .... k. This sample space than consists of a collection of subspaces 80. each of the type «ulcerated by some H(d‘. 31o .... ‘1‘: 1:)- If the previous instructions are altered slightly we obtain a 52 different smnple space, issue we would like to divide the group into categories containing unequal numbers of members. For instance. the group may have 1: projects to complete simultaneously and the nth project requires v.“1 members. The instructions request each member to choose v‘ others for the uth project. The marginal row total vectors are then of the form 5“ = ('u- v‘. .... v‘). u = l. 2. .... k. where “i=1 v‘ é n-l is the only restriction on the vu's. The sanplc space determined by these instructions is the one consisting of all those nxnxk matrices C having 3“ = (v... v‘. .... 'u)' u = l. 2. .... k. fliers the v.2L are a fixed set of positive integers such that $1 was n-l. le point out that if 'u = d for all u, then this ample space is equivalent to the preceding sample space. The case rcnarks now apply to sets of k n-dimensional vectors z‘. u = 1. 2. .... k. as in the preceding sample space. except. of course. that we no longer have ri‘ II 6.. for all i and a. le next consider a case in which the choices made by the members are unrestricted. In many situations mmberc of a group may be asked to olassib other members on a qualitative basis. That is. the criterion of the test may dcal with some individual trait such as loyalty or integrity; in such cases the categories would be characterised by such words as superior. excellent. good. etc. Classifications are then a matter of personal Judgment and the instructions would have to pemit unrestricted choices for the various categories. The marginal row total vectors are than random variables. The sample space is now the space of all nxnxk matrices C. For each set of marginal row and column total vectors (£1. 31. .... gt. 3:) we have a subspace Co of the sample space which is enmerated by 3(51. 11. .... 5k. 3‘). The total amber of nxnxk matrices o in this ample space is. by elementary combinatorial considerations. (k+l)n("1). Before proceeding to the hypothetical example we consider one more case. Assuno a sociometric test has been administered and the results recorded. This would result in a fixed matrix C with fixed c1 3‘. ri‘. and s1“. [a mu wish to confine attention to only those matrices having the sac fixed 5“ and g“. In this case. the appropriate smplo space would contain 8(51. _s}. .... f. 31‘) nxnxk matrices C. where the r“ and _s_“. u = l. 2. .... k. are those recorded in the test. It can be seen that the sample spaces to be used depend heavily upon the instructions given for the sociometric toot. Furthermore. the investigator may wish to alter his frame of reference by choosing one or other subset of all matrices c as his sample space. We next consider a particular example in which we show the complete process. including definition of the random variable to conform with the nature of the sociometric investigation. 24: A hypothetical example. Suppose we are interested in whether all. or most. or few of the members are "accepted" by the rest of the group. Let the sociometric test be one in which each individual is asked to make a first. a second and a third choice from among the other (n-l) individuals on the basis of some criterion. The (11-4) not selected by an individual are thus automatically classified in the fourth category by that individual. Let £1. 32. 33 be the n-dimen- sional marginal colunn total vectors representing the numbers of first. second and third Choices received. respectively. From the nature of the sociometric test uployed. we have rm 3 l for i = l. 2. .... n and u = l. 2. 3. It follows that the sample space consists of all those nxnx3 hollow matrices c with marginal row total vectors £1 = 53 = 53 = (l. l. .... l). |the total number of matrices in the sanple space can be found in two ways. First. from elanemtary combinatorial considerations. we see that the ith individual can make his first choice in (11-1) was. next. his second in (n—Z) ways. and. finally. his third in (n-3) wave; the remaining (n-4) choices are then determined. It follows that the total umber of ways is [(n-1)(n-3)(n-3)]n. since the n individuals make their choices independently. Second. the total number could be found by accumulating Hun. 31. 1‘. 3?. 1n. _s_3) for all possible distinct sets of colunn total vectors 3}. 32 and _s_3. (Since 5“ = (l. l. .... l) for u 2 l. 2. 3. this task could be simplified considerably by considering pemtations of the 'Ju’) Thus. the sample space contains [-(.n--l)(n--2)(n-L‘ofln nxnxz matrices c and for emeration purposes these can be classified into disjoint subspaces Go enmerated by EU“. _s_l. l‘. 32. 1‘. 33) for distinct sets of 3}. 32. 33. To each 55 of those matrices we assign a probability measure of one divided by En—l)(n-2)(n-3)1‘. This sanple space is the domain for the random variable x which must now be specified. Ie may choose to define acceptance of on individual by consider- ation of a weighted sum of choices. A weight of three is given to a first choice. a weight of two to a second choice. and a weight of one to a third choice. Thus. for the 1th individual we have a weighted sum of choices received. namely Sens!” 2'32+'33° Iron n _ . the nature of the responses we have 3:1 '31: = n. for each u = 1. 2. 3; therefore the expected value of .3“. for every 3 and u. is one. It follows that the expected value of the weighted sun for individual .1 is 6. 'e shall say that individual .1 is not fully accepted if his weighted sum is less than the expectation. Let x be the nunber of such individuals in a group of n indi- viduals. where individual a is not fully accepted if the weighted sun of his choices received is less than six. The domain of this random variable is the set of lin-l)(n--2)(n--'.’t))n nxnfl matrices c in the sample space. The‘range is the set of non-negative integers between 0 and (n-l). inclusive. One obvious way in which all members can be accepted (x = O) is '3‘ = 1. for all u and J. The value (n-l) is assumed by x in the following manner. Let any individual receive (n-l) first choices. he is obviously accepted. the remaining first choice and the n second and n third choices are easily distributed mag the other indivichnls so that weighted sum is five or less; this is true only if n )4. The single valued. measure preserving mapping is evident by inspection. Given any nxnxk matrix C in the domain (sample space) we can determine the value of x for this matrix by examining. for each .1. 351+ 232+ ads and take x to be the number of .1 for which this weighted sum is less than six. This is a many to one mapping. hr each X in the range space. 0 $15 n-l, (n34). there is a class of 11113 matrices in the sanple space. The probability that x = x. is the probability of the inverse image of x in the domain. Thus. the probability distribution in the range space is induced by the single valued measure preserving mapping. In actual practice it would be necessary to find 3(1‘. 3}. l‘. _s_z. 1‘. _s_3) for all distinct sets of three n-dimensional marginal column total vectors 31. _s_2 and 33- The random variable x assures a value for each set. This procedure leads to the probability distribution of the random variable 1:. Thus. once the universe of discourse has been decided upon. many random variables can be defined by procedures similar to that given above. In closing we remark that meaningful random variables can best be selected through the cooperation of sociologists and mathematical statisticians. The inmediate future applications of the results of this thesis will probably be in this field. SUM! The problem considered in this paper was the extension of a result originally obtained by Kate and Powell [5]. They were con- corned with the one dimensional theory of group organisation as a complex of irrefloxive binary relationships. taking values of 0 and 1. between the pairs of individuals. In this paper the binary re- striction is removed and replaced by (k + l)-na.ry relationships. k > 1. The problem was solved in three successive stages. The successive stages will be found in the appendix. section 1.5. and section 1.6. respectively. A hypothetical ox-ple which dcmcnstrates the appli- cation of the main theorem is given in Chapter 2. The first two stages are special cases of the third stage. although the methods of proof in each of the three stages are unique. It was considered important to indicate the evolution of the process which led to the general result. It is doubtful that the general result would have been attained so soon by a direct attack. The success at each stage not only served as a natural impeller to the next stage. but also provided clues for the method of attack in the following stage. The three stages are described below. b l) Ternary relationships between ordered pairs of distinct indi- viduals; each individual chooses exactly one for category one. exactly one other for category two. and the remaining n-z are automatically classified in the third category. 58 2) Ternary relationships between ordered pairs of distinct indi- viduals; each individual classifies each of the others into one and only one of the three categories. 3) (k+ l)nary relationships (k > 1) between ordered pairs of distinct individuals; each individual classifies each of the others into one and only one of the (k-t-l) categories. These relationships between ordered pairs of individuals have matrix representations. this method of representation being introduced by Iersyth and sets [a] in 1946. In all three stages the matrices are three dinensional. nucly nxnxk for groups of n—individsals and (k't' 1) relationships. The third dimension of the matrix. 1:. is always one loss than the number of relations because of the (kit 1)st relationship (null relationship) which is mandatory for every ordered pair not having one of the other k relationships. The number of distinct matrices satisfying the restrictions is given by 3(51. 3}. £2. _s_z. .... gt. _s_k). where 5‘ and _s_“ are n-dinensienal vectors. In relation to the result of Kate and Powell [5]. B(r.g) = 16L». The first stage results in a theorcn which gives Il(l‘.s1 .9513) as a bilinoar function of «(1'23 1)ovl‘. 32) ). This proof. found in the appendix. centers on the fact that 1131-2 = (l. l. .... 1). hence there can be at nest one violation of the restriction that one and only one relationship exists between each ordered pair. A perm- tation function of row and solicit indicoe is the distinctive feature of the pnef. The second stage. section 1.5. results in a theorem mica expresses 59 3(51. 3}. £3. 33) as a bilinoar function of «(51. 31) «(52. 33). This proof has the sane feature as the proof in the appendix. nanely there is only one m to violate the restriction that one and only one rela- tion exists between each ordered pair. However. in this case there can be one or more violations in a row. Introduction of a new function. Eigm,3m§u(£1. 31. £2. 32) circmvents this difficulty. The third and last stage. section 1.6. expresses 3(51. 9}. £2. 32. .... _rF. _s_“) as a function of 9‘51. 3}) ’UEZ’ £2)"'Vl(£k. gt). The essential difference between this and the preceding cases is the restriction that one and only one relationship exists between each ordered pair can be violated in 2k - k - 1 ways for each ordered pair of distinct individuals. The machinery used in section 1.5 is adapted to handle this case. In each of the above stages. 3(51. 9}. 52. _s_z. .... 3“. 3k) is also expressed as a function of Sukhatme's [8] bipartitional functions My}. 31mg”. gay-Mg". 3“). This is made possible by rate and Powell's [5] theoras which expresses To; . _s_) as a function of “5.3). Ithis is very fortunate indeed since “5.3) is tabulated and published for partitions up throuai eight in Sukhatme [8] and for partitions up throuai twelve in David and Kendall [1]. It is extremely doubtful that tables of 15.3) will ever be published except for very special cases. in particular. cases where _r_ is of the fem (d.d... ..d). where d is a mall positive integer. One reason for this is that the ordering in the n—dimensional vectors _r_ and _s_ is important. and this fact alone would tend to make excessively long 60 tables. Tables are being constructed at the present time for «(1&9 for values of a up through 16. These tables are being constructed under the direction of Kate and. if published. will be useful for any cases involving 3(1‘. 31. In. 33. .... 1‘. 3k). i.e. 3:1 3 £2 2 3:3 = =3? = (l. l. .... 1). In relation to sociometric testing the extension can be beneficial. Prior to this extension the application to sociometric testing was confined to criteria which had only two categories of response. lJ'.'ho extension is applicable to any number of responses between individuals on the basis of a single criterion. There could be. for example. a criterion in which each manber was to rank. partly or completely. the remaining members. The method is also adaptable to criteria in which the responses have one or more scalar values. for fixed n and I: the total sanple space of all nxnxk matrices. satisfying the given restrictions. contains (k+1)n(n-l) distinct matrices. Althougi random variables can be defined on this space the more interesting sample spaces are subspaces of this space. These subspaces become the universes of discourse for defining random variables which are single valued. measure preserving mappings from the domain to range space of the variable. the range space being a subset of the real line. Given any sunple space there is a large class of random variables which could be defined. It is not the purpose of this paper to define these random variables. However. exact distributions can be obtained in man interesting cases by uploying the results of this paper. we 61 say this in a rather light vein; the actual task of computation. even for small n and. k. would be a long exacting process. though not an impossible tank. The mechanics of constructing a random.variablo. given a universe of discourse. are discussed.in Chapter 2.. A.hypothetical example is given as an illustration. Prior to concluding remarks. we point out that the results of this study may be adapted to problems in comnunications networks. ; simple example occurs in the onuseration of (or testing whether there exist any) possible networks which may be formed among stations con- taining specified nunbers of several kinds of outgoing and incoming trunk lines. In conclusion. this study has developed a probability model for group organisation theory with multi-valued relations between persons. It is now possible to obtain exact probability distributions for meaningful random.variables. The immediate future application of the results would appear to be in the field of sociometry. This will require close cooperation between sociologists and.mathomatioa1 statisticians. m1! A.1 Preliminaries The first successful attempt to extend the counting theorem of Kat: and Powell [5] from binary to ternary response is given below. In the terminology of section 1.1 the case under consideration is Run. _1. In. _s_z) where the row totals for categories one and two are all one. It is only natural that the first case emined would be an especially simple case of three categories of response. This special case is applicable to sociometric tests of the following type. Bach individual in a group of n individuals 222; classify exactly one individual in category one. exactly one individual in category too. and the remaining (ti-3) individuals into category three. In the corresponding matrix representation (using notation of Section 1.1) it follows that for fixed 1. c131 2 l for exactly one .1 (corresponding to category one) . c132 = l for exactly one other J (corresponding to category two). and the remaining (n—3) positions in the ith row have c131 = c1.13 = 0. By convention °iil = c112 0 for all i. This implies r11 = r12 = l for all 1. since ”in = écuu for each u = l. 2. From the mandatory restriction given by r:l = (1'11. r21. .... rn)=_1;2 = (r12. r23. .... rnz) = (l. l. .... 1) it follows that _s_1 2 (on. em. .... 'nl) and 32 = (on. egg. .... 'nz) are n-part. non-negative. ordered partitions of n (t1 = t2 = n in the 63 notation of section 1.1). Thus. in the nxnxz matrices c : (gun), we have cu“ = O or l (121:); 1.3 = l. 2. .... n; u = l. 2). and ciiu E O for all i and u. Associated with each matrix 0 are n—dimen- sional marginal total vectors (in. :1. ln. 32); we have denoted 51 g 3 = (1. 1, 1) by 1“. Since each individual classifies each of the others into one and only one of the three categories. it follows that. in the matrix representation. we have the restriction c1514.- c112 = 0 or 1 for every ordered pair (1.3) of distinct individuals. in attempt was made to find a related function. similar to one of sukhatme's [8] bipartitional functions, to cope with the matrices defined above. This effort was fruitless. However. it was realised that if the n-dimensional marginal total vectors (1". _s_l) and (in. 32) were considered independently. then. using the methods of Kate and Powell [5]. there existed two spaces of nxn matrices c consisting of 11“, 31) and Tl“. _s_a) matrices. respectively. Furthermore. the product space 6 of these two spaces contained a proper subspace consisting of all the matrices satisfying the given sociometric test. However. in general there were many instances in which c1 31+ c132 = 2 for one or more ordered pairs (1..” and such matrices violate the restriction at.” + C132 3 0 or 1. These matrices were referred to as coincidence matrices since. at one or more positions (1.1). there were two ones coinciding. The problem then was to algebraically decom- pose tho product space 8 in such a manner that the required matrices —n—‘.JTI'H Y“ could be counted. In order to do this we had to nploy operators J 61 which we now define. .1 Add 61(1'11' .... r11. .... rm. on. .... s31. .... ’nl' r13. eee, r12, eee, rig. .12, eee, .32, eee. .n2) = (r11. eee. til-1’ eee, rhl. C11. eee. .Jl-l. 0°09 Inlo r129 eeeo riz'lo 00's rigs .12. -°°O .Jz-li "" .n2)e The effect of the operator (Si is to replace the four n-part. non-negative. ordered partitions of n by four n-part. non-negative. ordered partitions of (n-l). That is. the operator 6? simultaneously reduces the ith component of 51 and 53 and simultaneously re- duces the Jth component of _s_]- and 33 each by unity. Thus. 6? operates simultaneously on two pairs of n-dimonsional marginal row and column total vectors (£1. _s_l) and (£2. _s_z) in precisely the one manner as the (‘11 introduced by Kate and Powell [57 operates on one pair of n-dimensional marginal row and column total vectors (5. g). It is easily shown that if (5. _s) (in section 4 of refer- ence [5]) is replaced by (51. 31. 52. _s_z) and if 6i is defined as in 51.1 above. then the latter 6'11 has all the properties of the 6": introduced by Kat: and Powell [5]. In particular. we desire to use the inverse of (1+ 6%) which they proved was the algebraic inverse. lo state this in the form of a lama. Lemma ad The operator (1+ 631) has an inverse. left and right. given by 65 1.1.2 (1+ girl z 1 - 5f... ((5.11); .. (riff-*- . where 6: is the operator defined by 1.1.1. There is a very special feature of the product space 6 around which the proofs are formulated. This feature is that r11 3 r12 = l for all i = l. 2. .... n. Hence. in the product space 8 . there is at most one coincidence (c131 = c132 = l) in each row. rurthemore. Sin". 31. 1". _e_3) reduces both r11 and 1-12 to sore and the resulting matrix can have no ones in the ith row. Iith regard to coincidences. it is also quite obvious that the maxim umber of coincidences in 3th column is equal to the minimum of '31 and s.122 . Io shall have occasion to refer to this and therefore we make the following definition. Lot of = min(sn.o32) andlet _s_. =(s1.o '2" .... sn‘). It follows that 3" gives the maxinun number of coincidences in each of the columns . Since there is at most one coincidence in each row of the matrices in the product space 8 we can decompose this space on this basis. Definition n.l Let 3- be the subspace of the product space 8 which has coincidences (ct.11 = cu2 = l) in exactly m rows. Using this definition it follows that C : 80+ 61+ 82+ . and that this decomposition is into mutually exclusive and exhaustive sub spaces. 66 d.2 Method of attack. Bach matrix in the product space 0 can be expressed as the sum of two matrices A and B. The matrices a will contain only coincidences and the matrices B will contain no coincidences. For ample. if c is a matrix in the product space 8 having coincidences only in the rows 11 4 i8 4 (in. then A is a matrix with these respective coincidences and B is the matrix defined by c — A. in expression will be given which enmerates the number of matrices in each of the subspaces 8‘. m 2‘. O. This will involve a function 3(51. sl . £2. 32) which we now define. Dofinition A.2.1 Let 3“}. 219 3o 33) be the tOt‘]. umber or distinct matrices in the produd space 8 which have ¢1J1+ c132 e or 1 for every ordered pair (1.3). 142:. Since an matrix which has m coincidences must have these to coincidences in m distinct rows we shall have frequent reference to subsets involving an arbitrary but specified number of rows. to now define such subsets and it should be noted that the definition has to do with row and column indices and not ordered pairs (1.5)- Definition 1.2.2 Let (imujm)ll denote an arbitrary but specified set of I distinct row indices i1 4 iz<---1 1 = 1. 2. see. n and ‘ g1 and 32 are an two n-part. non-negative. ordered partitions of n. then an exhaustive enumeration of the total nmnber of mm? matrices c in the product space 8 . by subspaces 811' is given by A.3.8 I10“. 31) rkfln. 32) '-' 1*J1T{i+ 6‘1}H(ln . _s_1 , ln. 32). Proof: Expanding the product on the right, we obtain A.3.4 'au‘. _s_l. l”. ng- 1;; 611m". 31. l“. 9.2) + 12:; 631532 3(1 _1 1 .32) + lithe ‘ The first term in A.3.4 is by definition the number of :1an matrices c in the subspace at) of the product space E . By the preceding Corollery A.3.2.l. the successive su-sations there- after give the total number in the subspaces e . ea. .... respec- tively. The theoran is an innnediate consequence. The primary objective was to find the amber of matrices C in the subspace Co. mely Hun. _s_l. 1n. _s_z). Is now employ Lemma a.l regarding inverse of (1+ 6f). In this special case r11 = r12 = l. for all 1. therefore (1+6g)'1 reduces to l - 6? + R. where all the tense in R are trivial since they involve (69m. m>l. TT 3 '1 : - Applying 14: .1 (11-61) 1“: (1 &£) to both members of A.3.3. we have inrnediately Theorem A.3.4 If r11 = r12 = l for all i = 1. 2. ..., n and s1 and 32 are any two n-part. non-negative. ordered partitions of n. then the total number of nxnxz matrices c in the subspace GO of the product space 6 . is given by A.8.5. nun, 31.1“ ”mg - 5}y\(l‘ . s1)r\(ln,s where 6% operates simultaneously on (1n. 3}) and (1n, _s_z le note. in closing, that this is a special case of Theorem 1.5.4. namely r11 = r13 3 l for all i, but remark that it was included because of the distinctness of the proof. 73 W033 Dafid. LN. and LG. Kendall. “Tables of symmetric functions. Parts II and III.“ Biometrika. Vol.38 (1951). pp. 435-462. roller. I. An Introduction to Probability Theory and Its Appli- cations. John Iilq and Sons. New York. 1950. Persyth. E. and 1.. Rate. 'A matrix approach to the analysis of sociometric data: Preliminary report.” Sociomotg. Vol. 9 (1946). pp. 340-347. late. 1.. I'oil the matric analysis of sociometric data.” Sociemetg. Vol. 10 (1947). pp. 233-240. Eats. L. and JJI. Powell. "The nmsber of locally restricted directed graphs.‘ Proc. Amer. Math.fi§ee.. Vol. 5 (1954). We 621-626o Moreno. J .L. the Shall Surviveh A [gwgproach to the Problem of mason Interrelationg. Beacon House. New York. 1934. Powell. J .H. "A mathematical model for single function group organisation theory with applications to sociometric investi— gations," _Thgsie for the Degree of Ph. D.. Iiehigan State College. 1954. sukhatme. P. Y. “on bipartitienal functions." Philos. Trans. Roz. soc. London. Ser. A. Vol. 237 (1938). pp. 375-409. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII LLLLILLLILLLLLLLLLLLLLLLLLLLL LL L||LL LL LII LLLLLLLLLLLLL 51