Narayana numbers and Schur–Szegő composition

Abstract

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 = 1 n C n k − 1 C 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–Szegő composition map sending ( n − 1 ) -tuples of polynomials of the form ( x + 1 ) n − 1 ( x + a ) to their Schur–Szegő 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 ) } .