25 research outputs found
Elliptic rook and file numbers
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers
for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's
q-rook numbers by two additional independent parameters a and b, and a nome p.
These are shown to satisfy an elliptic extension of a factorization theorem
which in the classical case was established by Goldman, Joichi and White and
later was extended to the q-case by Garsia and Remmel. We obtain similar
results for our elliptic analogues of Garsia and Remmel's q-file numbers for
skyline boards. We also provide an elliptic extension of the j-attacking model
introduced by Remmel and Wachs. Various applications of our results include
elliptic analogues of (generalized) Stirling numbers of the first and second
kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings
has been taken out to keep the length of this paper reasonable
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
-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
The combinatorics of Jeff Remmel
We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics
Eulerian polynomials via the Weyl algebra action
Through the action of the Weyl algebra on the geometric series, we establish a generalization
of the Worpitzky identity and new recursive formulae for a family of
polynomials including the classical Eulerian polynomials. We obtain an extension
of the Dobi´nski formula for the sum of rook numbers of a Young diagram by replacing
the geometric series with the exponential series. Also, by replacing the derivative
operator with the q-derivative operator, we extend these results to the q-analogue setting
including the q-hit numbers. Finally, a combinatorial description and a proof of
the symmetry of a family of polynomials introduced by one of the authors are provided
\u3cem\u3eq\u3c/em\u3e-Stirling Identities Revisited
We give combinatorial proofs of q-Stirling identities using restricted growth words. This includes a poset theoretic proof of Carlitz\u27s identity, a new proof of the q-Frobenius identity of Garsia and Remmel and of Ehrenborg\u27s Hankel q-Stirling determinantal identity. We also develop a two parameter generalization to unify identities of Mercier and include a symmetric function version