42 research outputs found
Symmetric unimodal expansions of excedances in colored permutations
We consider several generalizations of the classical -positivity of
Eulerian polynomials (and their derangement analogues) using generating
functions and combinatorial theory of continued fractions. For the symmetric
group, we prove an expansion formula for inversions and excedances as well as a
similar expansion for derangements. We also prove the -positivity for
Eulerian polynomials for derangements of type . More general expansion
formulae are also given for Eulerian polynomials for -colored derangements.
Our results answer and generalize several recent open problems in the
literature.Comment: 27 pages, 10 figure
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J.
Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the
coefficients in an expansion of generalized Eulerian polynomials, which implies
the symmetric and unimodal property of the Eulerian numbers. We show that such
a formula with the conjectured property can be derived from the combinatorial
theory of continued fractions. We also discuss an analogous expansion for the
corresponding formula for derangements and prove a -analogue of the fact
that the (-1)-evaluation of the enumerator polynomials of permutations (resp.
derangements) by the number of excedances gives rise to tangent numbers (resp.
secant numbers). The -analogue unifies and generalizes our recent
results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that
of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).Comment: 19 pages, 2 figure
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials
We propose a new approach to the combinatorial interpretations of
linearization coefficient problem of orthogonal polynomials. We first establish
a difference system and then solve it combinatorially and analytically using
the method of separation of variables. We illustrate our approach by applying
it to determine the number of perfect matchings, derangements, and other
weighted permutation problems. The separation of variables technique naturally
leads to integral representations of combinatorial numbers where the integrand
contains a product of one or more types of orthogonal polynomials. This also
establishes the positivity of such integrals.Comment: Journal of Combinatorial Theory, Series A 120 (2013) 561--59
Cyclic derangements
A classic problem in enumerative combinatorics is to count the number of
derangements, that is, permutations with no fixed point. Inspired by a recent
generalization to facet derangements of the hypercube by Gordon and McMahon, we
generalize this problem to enumerating derangements in the wreath product of
any finite cyclic group with the symmetric group. We also give q- and (q,
t)-analogs for cyclic derangements, generalizing results of Brenti and Gessel.Comment: 14 page
The Combinatorics of Al-Salam-Chihara -Laguerre polynomials
We describe various aspects of the Al-Salam-Chihara -Laguerre polynomials.
These include combinatorial descriptions of the polynomials, the moments, the
orthogonality relation and a combinatorial interpretation of the linearization
coefficients. It is remarkable that the corresponding moment sequence appears
also in the recent work of Postnikov and Williams on enumeration of totally
positive Grassmann cells.Comment: 23 pages, to appear in Adv. in Appl. Math
Some identities on derangement and degenerate derangement polynomials
In combinatorics, a derangement is a permutation that has no fixed points.
The number of derangements of an n-element set is called the n-th derangement
number. In this paper, as natural companions to derangement numbers and
degenerate versions of the companions we introduce derangement polynomials and
degenerate derangement polynomials. We give some of their properties,
recurrence relations and identities for those polynomials which are related to
some special numbers and polynomials.Comment: 12 page