"Define a permutation to be any sequence of distinct positive integers. Given two permutations pi and sigma on disjoint underlying sets, we denote by pi shuffle sigma the set of shuffles of pi and sigma, that is, the set of all permutations obtained by interleaving the two permutations. A permutation statistic is a function, St, whose domain is the set of permutations and has the property that St(pi) only depends on the relative order of the elements of pi. A permutation statistic is shuffle compatible if the distribution of St on pi shuffle sigma depends only on the lengths of pi and sigma and St(pi) and St(sigma) rather than on the individual permutations themselves. This notion is implicit in the work of Stanley when he developed his theory of P-partitions, where P is a partially ordered set. The definition was explicitly given by Gessel and Zhuang who proved that various permutation statistics were shuffle compatible using mainly algebraic means. This work was continued by Grinberg. The purpose of the present work is to use bijective techniques to give demonstrations of shuffle compatibility. In particular, we show how a large number of permutation statistics can be shown to be shuffle compatible using a few simple bijections. Our approach also leads to a method for constructing such bijective proofs rather than having to treat each one in an ad hoc manner. Finally, we are able to prove a conjecture of Gessel and Zhuang about the shuffle compatibility of a certain statistic."--Page ii.
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