Orbits of antichains in certain root posets

TL;DR

This paper provides an alternative proof for the average antichain size in certain posets, extends the result to new posets, and suggests a broader root poset framework for the homomesy phenomenon.

Contribution

It introduces a new proof method for antichain averages and extends the results to additional root posets, proposing a unified root poset setting for homomesy.

Findings

Average antichain size in $[m]\times [n]$ orbits is $\frac{mn}{m+n}$

Average size in $[m]\times K_{n-1}$ orbits is $\frac{2mn}{m+2n-1}$

Root posets may unify the understanding of homomesy phenomenon

Abstract

This paper gives another proof of Propp and Roby's theorem saying that the average antichain size in any reverse operator orbit of the poset $[m]\times [n]$ is $\frac{mn}{m+n}$. It is conceivable that our method should work for other situations. As a demonstration, we show that the average size of antichains in any reverse operator orbit of $[m]\times K_{n-1}$ equals $\frac{2mn}{m+2n-1}$. Here $K_{n-1}$ is the minuscule poset $[n-1]\oplus ([1] \sqcup [1]) \oplus [n-1]$. Note that $[m]\times [n]$ and $[m]\times K_{n-1}$ can be interpreted as sub-families of certain root posets. We guess these root posets should provide a unified setting to exhibit the homomesy phenomenon defined by Propp and Roby.