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}

There and Back Again

We present a programming pattern where a recursive function traverses a data structure--typically a list--at return time. The idea is that the recursive calls get us there (typically to a base case) and the returns get us back again while traversing the data structure. We name this programming pattern of traversing a data structure at return time ``There And Back Again'' (TABA). The TABA pattern directly applies to computing a symbolic convolution. It also synergizes well with other programming patterns, e.g., dynamic programming and traversing a list at double speed. We illustrate TABA and dynamic programming with Catalan numbers. We illustrate TABA and traversing a list at double speed with palindromes and we obtain a novel solution to this traditional exercise. A TABA-based function written in direct style makes full use of an Algol-like control stack and needs no heap allocation. Conversely, in a TABA-based function written in continuation-passing style, the continuation acts as a list iterator. In general, the TABA pattern saves one from constructing intermediate lists in reverse order.See also BRICS-RS-05-3

Bijective Recurrences for Motzkin Paths

Consider lattice paths on Z² with steps (1; 1), (1; 1), and (1; 0). For n 2, let E n denote the set of such paths running from (0; 0) to (n; 0) and remaining strictly above the x-axis except initially and terminally. The cardinalities, f n = jE n j, are the Motzkin numbers, 1; 1; 2; 4; 9; 21; 51; 127; : : :, for n 2. We define a bijection yielding the recurrence (n + 1)f n+1 = (2n 1)f n + 3(n 2)f n 1 , for n 3. A modification of the bijection proves that the sum of the areas under the paths of E n , denoted by A n , satises A n+1 = 2A n +3A n 1 ; for n 3. A second modification yields a recurrence for a second moment on E n which agrees with Euler's recurrence for the central trinomial numbers

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

Journal of Integer Sequences, Vol. 6 (2003), Article 03.1.5 OBJECTS COUNTED BY THE CENTRAL DELANNOY NUMBERS

(A001850 of The On-Line Encyclopedia of Integer Sequences) will be defined so that dn counts the lattice paths running from (0, 0) to (n, n) that use the steps (1, 0), (0, 1), and (1, 1). In a recreational spirit we give a collection of 29 configurations that these numbers count. 1

Objects Counted By The Central Delannoy Numbers

The central Delannoy numbers, (d n ) n0 = 1; 3; 13; 63; 321; 1683; 8989; 48639; : : : (A001850 of The On-Line Encyclopedia of Integer Sequences) will be defined so that dn counts the lattice paths running from (0; 0) to (n; n) that use the steps (1; 0), (0; 1), and (1; 1). In a recreational spirit we give a collection of 29 configurations that these numbers count

Counting Lattice Paths By Narayana Polynomials

Let d(n) count the lattice paths from (0, 0) to (n, n) using the steps (0,1), (1,0), and (1,1). Let e(n) count the lattice paths from (0, 0) to (n, n) with permitted steps from the step set NN f(0; 0)g, where N denotes the nonnegative integers. We give a bijective proof of the identity e(n) = 2 n 1 d(n) for n 1. In giving perspective for our proof, we consider bijections between sets of lattice paths defined on various sets of permitted steps which yield path counts related to the Narayana polynomials

Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.1 Moments of Generalized Motzkin Paths

Abstract: Consider lattice paths in the plane allowing the steps (1,1), (1,-1), and (w,0), for some nonnegative integer w. For n> 1, let E(n,0) denote the set of paths from (0,0) to (n,0) running strictly above the x-axis except initially and finally. Generating functions are given for sums of moments of the ordinates of the lattice points on the paths in E(n,0). In particular, recurrencess are derived for the cardinality, the sum of the first moments (essentially the area), and the sum of the second moments for paths in E(n,0). These recurrences unify known results for w = 0, 1, 2, i.e. those for the Dyck (or Catalan), Motzkin, and Schröder paths, respectively. The sum of the second moments is seen to equal the number of unrestricted paths running from (0,0) to (0,n-2). Contents