A Combinatorial Proof of the Recurrence for Rook Paths

Let an count the number of 2-dimensional rook paths Rn,n from (0, 0) to (2n, 0). Rook pathsRm,n are the lattice paths from (0, 0) to (m+n,m−n) with allowed steps (x, x) and (y,−y) where x, y ∈ N+. In answer to the open question proposed by M. Erickson et al. (2010), we shall present a combinatorial proof for the recurrence of an, i.e., (n + 1)an+1 + 9(n − 1)an−1 = 2(5n + 2)an with initial conditions a0 = 1 and a1 = 2. Furthermore, our proof can be extended to show the recurrence for the number of multiple Dyck paths dn, i.e., (n+ 2)dn+1 + 9(n− 1)dn−1 = 5(2n+ 1)dn with d0 = 1 and d1 = 1, where dn = Nn(4) and Nn(x) is Narayana polynomial.