Descents of Permutations in a Ferrers Board.

The classical Eulerian polynomials are defined by setting A(n)(t) = Sigma(sigma is an element of Gn) t(1+des(sigma)) = Sigma(n)(k=1) A(n,k)t(k) where A(n,k) is the number of permutations of length n with k - 1 descents. Let A(n)(t, q) = Sigma(pi is an element of Gn) t(1+des(pi))q(inv(pi)) be the inv q-analogue of the classical Eulerian polynomials whose generating function is well known: Sigma(n >= 0) u(n)A(n)(t, q)/[n](q)! = 1/1-t Sigma(k >= 1)(1-t)(k)u(k)/[k](q)! (0.1) In this paper we consider permutations restricted in a Ferrers board and study their descent polynomials. For a general Ferrers board F, we derive a formula in the form of permanent for the restricted q-Eulerian polynomial A(F)(t, q) := Sigma(sigma is an element of F)t(1+des(sigma))q(inv)(sigma). If the Ferrers board has the special shape of an n x n square with a triangular board of size s removed, we prove that A(F)(t, q) is a sum of s + 1 terms, each satisfying an equation that is similar to (0.1). Then we apply our results to permutations with bounded drop (or excedance) size, for which the descent polynomial was first studied by Chung et al. (European J. Combin., 31(7) (2010): 1853-1867). Our method presents an alternative approach.