Counting descent pairs with prescribed tops and bottoms

Given sets X and Y of positive integers and a permutation σ = σ 1 σ 2 ⋯ σ n ∈ S n , an ( X , Y ) -descent of σ is a descent pair σ i > σ i + 1 whose “top” σ i is in X and whose “bottom” σ i + 1 is in Y. We give two formulas for the number P n , s X , Y of σ ∈ 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 mod k , math.CO/0604455] on counting descent pairs whose top (or bottom) is equal to 0 mod k .