Three dimensional Narayana and Schröder numbers

AbstractConsider the 3-dimensional lattice paths running from (0,0,0) to (n,n,n), constrained to the region {(x,y,z):0⩽x⩽y⩽z}, and using various step sets. With C(3,n) denoting the set of constrained paths using the steps X≔(1,0,0), Y≔(0,1,0), and Z≔(0,0,1), we consider the statistic counting descents on a path P=p1p2…p3n∈C(3,n), i.e., des(P)≔|{i:pipi+1∈{YX,ZX,ZY},1⩽i⩽3n-1}|. A combinatorial cancellation argument and a result of MacMahon yield a formula for the 3-Narayana number, N(3,n,k)≔|{P∈C(3,n):des(P)=k+2}|. We define other statistics distributed by the 3-Narayana number and show that 4∑k2kN(3,n,k) yields the nth large 3-Schröder number which counts the constrained paths using the seven positive steps of the form (ξ1,ξ2,ξ3), ξi∈{0,1}

Generalizing Narayana and Schröder numbers to higher dimensions

Let C(d, n) denotethesetofd-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}. Onanypath P: = p1p2...pdn ∈C(d, n), define the statistics asc(P):=|{i: pipi+1 = XjXℓ,j < ℓ} | and des(P):=|{i: pipi+1 = XjXℓ,j> ℓ}|. Define the generalized Narayana number N(d, n, k) tocountthepathsinC(d, n) withasc(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 k N(d, n, k)2k)n≥1 to count constrained paths using step sets which include diagonal steps