Noncrossing partitions, toggles, and homomesies

TL;DR

This paper introduces involutions called toggles on noncrossing partitions and demonstrates that many functions, including block count, have constant orbit averages (homomesy) under these toggles, with broader applications to graph independent sets.

Contribution

It develops a new framework of toggle involutions on noncrossing partitions and extends homomesy results to these and related graph structures, advancing dynamic algebraic combinatorics.

Findings

Homomesy holds for many functions under toggle operations on noncrossing partitions.

The toggle method applies to independent sets in certain graphs, including '2-cliquish' graphs.

The approach generalizes to broader classes of combinatorial objects and operations.

Abstract

We introduce $n(n-1)/2$ natural involutions ("toggles") on the set $S$ of noncrossing partitions $\pi$ of size $n$, along with certain composite operations obtained by composing these involutions. We show that for many operations $T$ of this kind, a surprisingly large family of functions $f$ on $S$ (including the function that sends $\pi$ to the number of blocks of $\pi$) exhibits the homomesy phenomenon: the average of $f$ over the elements of a $T$-orbit is the same for all $T$-orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "$2$-cliquish". More generally, the philosophy of this "toggle-action", proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.