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 =

Interlacing properties and the Schur-Szegő composition

“…Remark 21 It is shown in paper [9] (see part 1) of Corollary 7 there) that the roots of the Narayana polynomials N n and N n−1 interlace, i.e. satisfy the first interlacing property.…”

“…3) It is shown in [9] that for j fixed and when n → ∞, the polynomial xQ j (−x) tends to the Narayana polynomial n i=1 N n,i x i where N n,i are the Narayana numbers C i n C i−1 n /n. Denote by 0 < x 1 < · · · < x j the roots of Q j , by 0 < y 1 < · · · < y j+1 the ones of Q j+1 and by 0 < z 1 < · · · < z j+2 the ones of Q j+2 .…”

Universal two-time correlations, out-of-time-ordered correlators, and Leggett-Garg inequality violation by edge Majorana fermion qubits

“…where u = µ 2∆ , and N m (x) = m n=1 N m,n x n are the well-known Narayana polynomials, which involve the Narayana numbers N m,n = 1 m m n−1 m n [47,48]. Note that the critical point corresponds to u = 1, for which…”

Dyck Words, Lattice Paths, and Abelian Borders

“…which can be seen by the fact that N n,r counts the number of Dyck paths of length 2n that have r peaks [19]. Using Equation (1), we get…”