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. aracterisation 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 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.