On counting permutations by pairs of congruence classes of major index

Barcelo, Helene; Maule, Robert; Sundaram, Sheila

On counting permutations by pairs of congruence classes of major index

Authors

Barcelo, Helene

Maule, Robert

Sundaram, Sheila

Description

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).
8 pages