q-Counting Descent Pairs with Prescribed Tops and Bottoms

Given sets X and Y of positive integers and a permutation @s=@s"1@s"2...@s"n@?S"n, an (X,Y)-descent of @s is a descent pair @s"i>@s"i"+"1 whose ''top''@s"i is in X and whose ''bottom''@s"i"+"1 is in Y. We give two formulas for the number P"n","s^X^,^Y of @s@?S"n with s(X,Y)-descents. P"n","s^X^,^Y is also shown to be a hit number of a certain Ferrers board. This work generalizes results of Kitaev and Remmel [S. Kitaev, J. Remmel, Classifying descents according to parity, math.CO/0508570; S. Kitaev, J. Remmel, Classifying descents according to equivalence modk, math.CO/0604455] on counting descent pairs whose top (or bottom) is equal to 0modk.