25 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
Generalized Derangements and Anagrams Without Fixed Letters
For a union of disjoint sets a permutation is a generalized derangement if no
element is mapped to an element in its own set. Denote the number of such
permutations by
For a given word denote by the number of anagrams of for which
no letter occupies a position which was occupied by the same letter in the
original word.
In this article we propose several new properties of the very closely related
functions and
After some definitions and preliminary observations, we proceed with two
recursive algorithms for computing and . We use the algorithms to prove
several inequalities which allow us to roughly estimate and partially order the
values of and
Finally, we turn to the number-theoretical properties of We prove three
theorems and propose four corollaries of the last of them. One of the results
in this section fully determines when is odd. The main approach in the
section is splitting the anagrams into classes of equivalence in different
ways.Comment: 19 pages; Comments welcome
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
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
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
Enumerative combinatorics, continued fractions and total positivity
Determining whether a given number is positive is a fundamental question in mathematics. This can sometimes be answered by showing that the number counts some collection of objects, and hence, must be positive. The work done in this dissertation is in the field of enumerative combinatorics, the branch of mathematics that deals with exact counting. We will consider several problems at the interface between
enumerative combinatorics, continued fractions and total positivity.
In our first contribution, we exhibit a lower-triangular matrix of polynomials in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. This generalises Brenti’s conjecture from 1996. We prove the coefficientwise total positivity of a three-variable case which includes the reversed Stirling subset triangle.
Our next contribution is the study of two sequences whose Stieltjes-type continued fraction coefficients grow quadratically; we study the Genocchi and median
Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations, a class of permutations of 2n, with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.
After this, we interpret the Foata–Zeilberger bijection in terms of Laguerre digraphs, which enables us to count cycles in permutations. Using this interpretation,
we obtain Jacobi-type continued fractions for multivariate polynomials enumerating permutations, and also Thron-type and Stieltjes-type continued fractions for multivariate polynomials enumerating D-permutations, in both cases including the counting of cycles. This enables us to prove some conjectured continued fractions due to Sokal–Zeng from 2022, and Randrianarivony–Zeng from 1996.
Finally, we introduce the higher-order Stirling cycle and subset numbers; these generalise the Stirling cycle and subset numbers, respectively. We introduce some
conjectures which involve different total-positivity questions for these triangular arrays and then answer some of them