Narayana numbers and Schur–Szegő composition
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...
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Veröffentlicht in: | Journal of approximation theory 2009-12, Vol.161 (2), p.464-476 |
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Sprache: | eng |
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Zusammenfassung: | 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
)
}
. |
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ISSN: | 0021-9045 1096-0430 1096-0430 |
DOI: | 10.1016/j.jat.2008.10.013 |