Generalizing Narayana and Schröder Numbers to Higher Dimensions

نویسنده

  • Robert A. Sulanke
چکیده

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 = XjX`, j < `}| and des(P ) :=|{i : pipi+1 = XjX`, j > `}|. 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 (WilfZeilberger) 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)2 )n≥1 to count constrained paths using step sets which include diagonal steps. Key phases: Lattice paths, Catalan numbers, Narayana numbers, Schröder numbers, Sister Celine’s (Wilf-Zeilberger) method Mathematics Subject Classification: 05A15

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عنوان ژورنال:
  • Electr. J. Comb.

دوره 11  شماره 

صفحات  -

تاریخ انتشار 2004