This thesis considers three algebraically motivated combinatorics questions on partially ordered sets (posets) and graphs. In the process, we consider rooted tree posets, inflated rooted tree posets, shoelace posets, $(3+1)$-free posets, as well as incomparability graphs of a given poset. Rooted trees are posets whose Hasse diagram is a graph-theoretic tree having a unique minimal element. We study rowmotion on antichains and lower order ideals of rooted trees. Recently Elizalde, Roby, Plante, and Sagan considered rowmotion on fences which are posets whose Hasse diagram is a path (but permitting any number of minimal elements). They showed that in this case, the orbits could be described in terms of tilings of a cylinder. They also defined a new notion called homometry where a statistic takes a constant value on all orbits of the same size. This is a weaker condition than the well-studied concept of homomesy which requires a constant value for the average of the statistic over all orbits. Rowmotion on fences is often homometric for certain statistics, but not homomesic. We introduce a tiling model for rowmotion on rooted trees. We use it to study various specific types of trees and show that they exhibit homometry, although not homomesy, for certain statistics.We also study Defant and Kravitz's generalization of Sch\"utzenberger's promotion operator to arbitrary labelings of finite posets. Defant and Kravitz showed that applying the promotion operator $n-1$ times to a labeling of a poset on $n$ elements always gives a natural labeling of the poset and called a labeling tangled if it requires the full $n-1$ promotions to reach a natural labeling. They also conjectured that there are at most $(n-1)!$ tangled labelings for any poset on $n$ elements. We propose a strengthening of their conjecture by partitioning tangled labelings according to the element labeled $n-1$ and prove that this stronger conjecture holds for inflated rooted forest posets and a new class of posets called shoelace posets. We also introduce sorting generating functions and cumulative generating functions for the number of labelings that require $k$ applications of the promotion operator to give a natural labeling. We prove that the coefficients of the cumulative generating function of the ordinal sum of antichains are log-concave and obtain a refinement of the weak order on the symmetric group.We also consider $(3+1)$-free posets, motivated by a reduction of the Stanley-Stembridge conjecture posited by Foley, Ho\`ang, and Merkel (2019), stating that the twinning operation on graphs preserves $e$-positivity of the chromatic symmetric function. A counterexample to this general conjecture was given by Li, Li, Wang, and Yang (2021). We prove that $e$-positivity is preserved by the twinning operation on cycles, by giving an $e$-positive generating function for the chromatic symmetric function, as well as an $e$-positive recurrence. We derive similar $e$-positive generating functions and recurrences for twins of paths. Our methods make use of the important triple deletion formulas of Orellana and Scott (2014), as well as new symmetric function identities.
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