q-Counting Descent Pairs with Prescribed Tops and Bottoms.

Given sets X and Y of positive integers and a permutation sigma = sigma(1)sigma(2) ... sigma(n) is an element of S(n), an (X,Y)-descent of sigma is a descent pair sigma(i) > sigma(i+1) whose "top" sigma(i) is in X and whose "bottom" sigma(i+1) is in Y. Recently Hall and Remmel [4] proved two formulas for the number P(n,s)(X,Y) of sigma is an element of S(n) with s (X,Y)-descents, which generalized Liese's results in [1]. We define a new statistic stat(X,Y)(sigma) on permutations sigma and define P(n,s)(X,Y) (q) to be the sum of q(statX,Y(sigma)) over all sigma is an element of S(n) with s (X,Y)-descents. We then show that there are natural q-analogues of the Hall-Remmel formulas for P(n,s)(X,Y) (q).