Antichain generating polynomials of posets

Abstract

This paper gives a formula for the antichain generating polynomial $\mathcal{N}_{[k]\times Q}$ of the poset $[k]\times Q$, where $[k]$ is an arbitrary chain and $Q$ is any finite graded poset. When $Q$ specializes to be a connected minuscule poet, which was classified by Proctor in 1984, we find that the polynomial $\mathcal{N}_{[k]\times Q}$ bears nice properties. For instance, we will recover the $B_n$-Narayana polynomial and the $D_{2n+2}$-Narayana polynomial. We collect evidence for the conjecture that whenever $\mathcal{N}_{[k]\times P}(x)$ is palindromic, it must be $\gamma$-positive. Moreover, the family $\mathcal{N}_{[2]\times [n]\times [m]}$ should be real-rooted and $\mathcal{N}_{[2]\times [n]\times [n+1]}$ should be $\gamma$-positive. We also conjecture that $\mathcal{N}_{Q}(x)$ is log-concave (thus unimodal) for any connected Peck poset $Q$.