70 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
Partitions of Matrix Spaces With an Application to -Rook Polynomials
We study the row-space partition and the pivot partition on the matrix space
. We show that both these partitions are reflexive
and that the row-space partition is self-dual. Moreover, using various
combinatorial methods, we explicitly compute the Krawtchouk coefficients
associated with these partitions. This establishes MacWilliams-type identities
for the row-space and pivot enumerators of linear rank-metric codes. We then
generalize the Singleton-like bound for rank-metric codes, and introduce two
new concepts of code extremality. Both of them generalize the notion of MRD
codes and are preserved by trace-duality. Moreover, codes that are extremal
according to either notion satisfy strong rigidity properties analogous to
those of MRD codes. As an application of our results to combinatorics, we give
closed formulas for the -rook polynomials associated with Ferrers diagram
boards. Moreover, we exploit connections between matrices over finite fields
and rook placements to prove that the number of matrices of rank over
supported on a Ferrers diagram is a polynomial in , whose
degree is strictly increasing in . Finally, we investigate the natural
analogues of the MacWilliams Extension Theorem for the rank, the row-space, and
the pivot partitions
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