Generalizing narayana and schröder numbers to higher dimensions

Let C(d, n) denote the set of d-dimensional lattice paths using the steps X₁ := (1, 0, . . . , 0), X₂ := (0, 1, . . . , 0), . . . , X_d := (0, 0, . . . , 1), running from (0, 0, . . . , 0) to (n,n, . . . , n), and lying in {(x₁, x₂, . . . , x_d) : 0 ≤ x₁ ≤ x₂ ≤ . . . x_d}. On any path P := _p1p2 ⋯ _pdn ∈ C(d, n), define the statistics asc(P) :=|{i : _pipi+1 = X_jX_ℓ,j < ℓ}| and des(P) :=|{i : _pipi+1 = X_jX_ℓ,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 (2^d-1∑_kN(d, n, k)2^k)n≥1 to count constrained paths using step sets which include diagonal steps.