Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Schur positivity and the q-log-convexity of the Narayana polynomials
Journal of Algebraic Combinatorics: An International Journal
| Hi-index | 0.00 |
In the present paper we find a new interpretation of Narayana polynomials N"n(x) which are the generating polynomials for the Narayana numbers N"n","k=1nC"n^k^-^1C"n^k where C"j^i stands for the usual binomial coefficient, i.e. C"j^i=j!i!(j-i)!. They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1-2) (2002) 311-326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [T. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67-82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909-2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147-3160]. Strangely enough Narayana polynomials also occur as limits as n-~ of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^n^-^1(x+a) to their Schur-Szego product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N"n(x)}.