Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id,π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id,π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups An and Bn. The expressions involve q-Stirling numbers of the second kind, and for the group An putting q = 1 yields the poly-Bernoulli numbers defined by Kaneko. © 2007 Elsevier Inc. All rights reserved.

• Type: Journal
• Year: 2007
• Where: JCT
• Authors: Jonas Sjöstrand
• Keywords: Bruhat Intervals, JCT 2007, Lower Bruhat Interval, Poincaré Polynomial