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.