Bruhat intervals as rooks on skew Ferrers boards

We characterise the permutations @p such that the elements in the closed lower Bruhat interval [id,@p] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations @p such that [id,@p] 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, and for the group A"n putting q=1 yields the poly-Bernoulli numbers defined by Kaneko.

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