Generalized q-Runyon numbers on truncated rectangle

Abstract

Fiirlinger and Hofbauer (J. Combin. Theory, Series A, 40(1985):248-264) showed the following combinatorial interpretation of q-analogs of Runyon numbers (or Narayana numbers) (l/n)(n/k)(k+1/2), "equ"which may be viewed as a refinement of the well known q-Catalan formula Σ_{p∈ Dn}q^maj(p)= 1/n+1[2n/n]. In this paper, we generalize the above q-analog of Runyon numbers in two perspectives: (i) the domain of the summation where the lattice paths are situated is changed from the northwest upper half of an n by n square to a rectangle with an isosceles triangle cut off from its southeast corner; (ii) the relative pair of statistics on a path (des, maj) now has several variations (stat, majstat), where stat is one of des, asc, flat, flat₀,fiat₁. Moreover, we prove a sequence of enumerative results for these statistics and their weighted version. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.