Abstract
Given a permutation $\tau$ defined on a set of combinatorial objects $S$, together with some statistic $f:S\rightarrow \mathbb{R}$, we say that the triple $\langle S, \tau,f \rangle$ exhibits homomesy if $f$ has the same average along all orbits of $\tau$ in $S$. This phenomenon was observed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). Propp and Roby studied homomesy in the set of order ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the order ideal. In this paper we extend their results to generalized rowmotion and promotion, together with a wider class of statistics in the product of two chains. Moreover, we derive similar results in other simply described posets. We believe that the framework we set up here can be fruitful in demonstrating homomesy results in order ideals of broader classes of posets.
Highlights
Consider a poset P, and let J(P) be the set containing all of the order ideals in P
Propp and Roby [14] were interested in studying the orbits these bijections introduce on J(P) and the statistics that are preserved along these orbits
(3) We introduce a set of functions such that any linear combination of them will constitute a homomesic static in our setting, and we observe that the functions studied by Propp and Roby can be constituted using linear combinations of these functions
Summary
Consider a poset P, and let J(P) be the set containing all of the order ideals in P. By Remark 2 any linear combination of the above functions is homomesic, Theorem 2.3 generalizes Theorem 2.2 by introducing a wide class of permutations and statistics whose triples with J(Qa,b) exhibit homomesy. After introducing these definitions, we show that there is a natural equivariant bijection between the set of order ideals under comotion and the set of increasing sequences under winching. We show that there is a natural equivariant bijection between the set of order ideals under comotion and the set of increasing sequences under winching
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