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 $[m]\times [n]$ 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.
@article{10_37236_7055,
author = {Chao-Ping Dong and Suijie Wang},
title = {Orbits of antichains in certain root posets},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {4},
doi = {10.37236/7055},
zbl = {1439.06003},
url = {http://geodesic.mathdoc.fr/articles/10.37236/7055/}
}
TY - JOUR
AU - Chao-Ping Dong
AU - Suijie Wang
TI - Orbits of antichains in certain root posets
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/7055/
DO - 10.37236/7055
ID - 10_37236_7055
ER -
%0 Journal Article
%A Chao-Ping Dong
%A Suijie Wang
%T Orbits of antichains in certain root posets
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/7055/
%R 10.37236/7055
%F 10_37236_7055
Chao-Ping Dong; Suijie Wang. Orbits of antichains in certain root posets. The electronic journal of combinatorics, Tome 24 (2017) no. 4. doi: 10.37236/7055
