Orbits of Antichains in Certain Root Posets

Building everything from scratch, we give another proof of Propp and Roby’s theorem saying that the average antichain size in any reverse operator orbit of the poset [m] × [n] is 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] ×Kn−1 equals 2mn m+2n−1 . Here Kn−1 is the minuscule poset [n − 1] ⊕ ([1] ⊔ [1]) ⊕ [n − 1]. Note that [m] × [n] and [m] × Kn−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.