A Combinatorial Proof of the Recurrence for Rook Paths

Abstract

Let $a_n$ count the number of $2$-dimensional rook paths $\mathcal{R}_{n,n}$ from $(0,0)$ to $(2n,0)$. Rook paths $\mathcal{R}_{m,n}$ are the lattice paths from $(0,0)$ to $(m+n,m-n)$ with allowed steps $(x,x)$ and $(y,-y)$ where $x,y\in\mathbb{N}^{+}$. In answer to the open question proposed by M. Erickson et al. (2010), we shall present a combinatorial proof for the recurrence of $a_n$, i.e., $(n+1)a_{n+1}+9(n-1)a_{n-1}=2(5n+2)a_n$ with initial conditions $a_0=1$ and $a_1=2$. Furthermore, our proof can be extended to show the recurrence for the number of multiple Dyck paths $d_n$, i.e., $(n+2)d_{n+1}+9(n-1)d_{n-1}=5(2n+1)d_n$ with $d_0=1$ and $d_1=1$, where $d_n=\mathcal{N}_n(4)$ and $\mathcal{N}_n(x)$ is Narayana polynomial.