Generalizing narayana and schröder numbers to higher dimensions

Abstract

Let C(d, n) denote the set of d-dimensional lattice paths using the steps X1 := (1, 0, . . . , 0), X2 := (0, 1, . . . , 0), . . . , Xd := (0, 0, . . . , 1), running from (0, 0, . . . , 0) to (n,n, . . . , n), and lying in {(x1, x2, . . . , xd) : 0 ≤ x1 ≤ x2 ≤ . . . xd}. On any path P := p1p2 ⋯ pdn ∈ C(d, n), define the statistics asc(P) :=|{i : pipi+1 = XjXℓ,j ℓ}|. Define the generalized Narayana number N(d, n, k) to count the paths in C(d, n) with 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 asc and 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 (2d-1∑kN(d, n, k)2k)n≥1 to count constrained paths using step sets which include diagonal steps.