Orbits of antichains in certain root posets (Q2411511)

Summary: Building everything from scratch, we give another proof of \textit{J. Propp} and \textit{T. Roby}'s [Electron. J. Comb. 22, No. 3, Research Paper P3.4, 29 p. (2015; Zbl 1319.05151)] 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 [loc. cit.].