On Counting Permutations by Pairs of Congruence Classes of Major Index

For a fixed positive integer n, let Sn denote the symmetric group of n! permutations on n symbols, and let maj(σ) denote the major index of a permutation σ. Fix positive integers k < l ≤ n, and nonnegative integers i, j. Let mn(i\k; j\l) denote the cardinality of the set {σ ∈ Sn : maj(σ) ≡ i mod k,maj(σ) ≡ j mod l}. In this paper we give some enumerative formulas for these numbers. When l divides (n− 1) and k divides n, we show that for all i, j, mn(i\k; j\l) = n! k · l .