Homomesy via toggleability statistics

Abstract

The rowmotion operator acting on the set of order ideals of a finite poset has been the focus of a significant amount of recent research. One of the major goals has been to exhibit homomesies: statistics that have the same average along every orbit of the action. We systematize a technique for proving that various statistics of interest are homomesic by writing these statistics as linear combinations of "toggleability statistics" (originally introduced by Striker) plus a constant. We show that this technique recaptures most of the known homomesies for the posets on which rowmotion has been most studied. We also show that the technique continues to work in modified contexts. For instance, this technique also yields homomesies for the piecewise-linear and birational extensions of rowmotion; furthermore, we introduce a q-analogue of rowmotion and show that the technique yields homomesies for "q-rowmotion" as well.

The birational Lalanne–Kreweras involution

“…We later learned that the h i homomesies for rowmotion were previously observed, conjecturally, by David Einstein. We remark that another way to prove the h i homomesies for rowmotion is to write them as a linear combination of "rook" statistics plus a linear combination of signed toggleability statistics, as discussed in [8] (see also [10]). Example 4.16.…”