Bruhat intervals as rooks on skew Ferrers boards

Details Bruhat intervals as rooks on skew Ferrers boards Bruhat intervals as rooks on skew Ferrers boards

2006-01-25

arxiv.org/abs/math/0601615v1

Mathematics

Combinatorics

16 pages, 9 figures

Scientific paper

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.

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