Narayana numbers and Schur–Szegő composition Original Research Article

In the present paper we find a new interpretation of Narayana polynomials Nn(x)Nn(x) which are the generating polynomials for the Narayana numbers View the MathML sourceNn,k=1nCnk−1Cnk where View the MathML sourceCji stands for the usual binomial coefficient, i.e. View the MathML sourceCji=j!i!(j−i)!. They count Dyck paths of length nn and with exactly kk 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→∞n→∞ of the sequences of eigenpolynomials of the Schur–Szegő composition map sending (n−1)(n−1)-tuples of polynomials of the form (x+1)n−1(x+a)(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 {Nn(x)}{Nn(x)}.