Generalizing Narayana and Schröder Numbers to Higher Dimensions

Let ${\cal C}(d,n)$ denote the set of $d$-dimensional lattice paths using the steps $X_1 := (1, 0, \ldots, 0),$ $ X_2 := (0, 1, \ldots, 0),$ $\ldots,$ $ X_d := (0,0, \ldots,1)$, running from $(0,\ldots,0)$ to $(n,\ldots,n)$, and lying in $\{(x_1,x_2, \ldots, x_d) : 0 \le x_1 \le x_2 \le \ldots \le x_d \}$. On any path $P:=p_1p_2 \ldots p_{dn} \in {\cal C}(d,n)$, define the statistics ${\rm asc}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j < \ell \}|$ and ${\rm des}(P) := $$|\{i : p_ip_{i+1} = X_jX_{\ell}, j>\ell \}|$. Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\cal C}(d,n)$ with ${\rm asc}(P)=k$. We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work. We examine other statistics for $N(d,n,k)$ and show that the statistics ${\rm asc}$ and ${\rm des}-d+1$ are equidistributed. We use Wegschaider's algorithm, extending Sister Celine's (Wilf-Zeilberger) method to multiple summation, to obtain recurrences for $N(3,n,k)$. We introduce the generalized large Schröder numbers $(2^{d-1}\sum_k N(d,n,k)2^k)_{n\ge1}$ to count constrained paths using step sets which include diagonal steps.