2006/01/25 by Jonas Sjöstrand, Sjostrand, Jonas
Computer Science · Mathematics · #05A15 #06A07 #14M15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0601615
openalex publication_date 2006/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 An and Bn. 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.