856 research outputs found
-Rook polynomials and matrices over finite fields
Connections between -rook polynomials and matrices over finite fields are
exploited to derive a new statistic for Garsia and Remmel's -hit polynomial.
Both this new statistic and another statistic for the -hit polynomial
recently introduced by Dworkin are shown to induce different multiset
Mahonian permutation statistics for any Ferrers board. In addition, for the
triangular boards they are shown to generate different families of
Euler-Mahonian statistics. For these boards the family includes Denert's
statistic , and gives a new proof of Foata and Zeilberger's Theorem that
is jointly distributed with . The family appears
to be new. A proof is also given that the -hit polynomials are symmetric and
unimodal
Automatic Enumeration of Generalized Menage Numbers
I describe an empirical-yet-rigorous, algorithm, based on Riordan's rook
polynomials and the so-called C-finite ansatz, fully implemented in the
accompanying Maple package
(http://www.math.rutgers.edu/~zeilberg/tokhniot/MENAGES ), MENAGES, that
reproduces in a few seconds, rigorously-proved enumeration theorems on
permutations with restricted positions, previously proved by quite a few
illustrious human mathematicians, and that can go far beyond any human
attempts.Comment: 15 pages. An extended version of the last of three invited talks
given by the author at the 71th Seminaire Lotharingien de Combinatoire, that
took place in Bertinoro, Italy, Sept. 16-18, 201
Bruhat intervals as rooks on skew Ferrers boards
We characterise the permutations pi such that the elements in the closed
lower Bruhat interval [id,pi] of the symmetric group correspond to non-taking
rook configurations on a skew Ferrers board. It turns out that these are
exactly the permutations pi such that [id,pi] corresponds to a flag manifold
defined by inclusions, studied by Gasharov and Reiner.
Our characterisation connects the Poincare polynomials (rank-generating
function) of Bruhat intervals with q-rook polynomials, and we are able to
compute the Poincare polynomial of some particularly interesting intervals in
the finite Weyl groups A_n and B_n. The expressions involve q-Stirling numbers
of the second kind.
As a by-product of our method, we present a new Stirling number identity
connected to both Bruhat intervals and the poly-Bernoulli numbers defined by
Kaneko.Comment: 16 pages, 9 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
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