Combinatorial aspects of the Delannoy Lattice

Abstract

The Delannoy numbers $d(n,k)$ count lattice paths from $(0,0)$ to $(n-k,k)$ using steps $(1,0),(0,1)$ and $(1,1)$. This paper introduces a graded poset $\mathcal{D}_n$ on the Delannoy paths ending on the line $x+y=n$, whose rank-generating function is the Delannoy polynomial $d_n(x)=\sum_{k=0}^n d(n,k)x^k$. We prove that $\\mathcal{D}_n$ is a self-dual lattice, which is call the Delannoy lattice. By establishing an explicit symmetric Boolean decomposition, we show that $\mathcal{D}_n$ is a symmetric Boolean order, thereby recovering the $\gamma$-positivity of $d_n(x)$. Such a decomposition is refined to a symmetric chain decomposition with the chain cover property, and is applied to determine all maximum antichains. We also investigate other combinatorial aspects of $\mathcal{D}_n$, including supersolvability, the M\"obius number, characteristic polynomials, and zeta polynomials.