An equipartition property for the distribution of multiset permutation inversions

The purpose of this paper is to present some enumerative results concerning the class F-k of permutations of the multiset {1(m1).2(m2),...,r(mr)} havi ng inversion number congruent to k module n, with n = m(1) + m(2) +... + m( r) and 0 less than or equal to k < n. We show that the enumeration of this family of permutations is connected to gcd(m(1), m(2),...,m(r)), and if gcd (m(1), m(2),...,m(r)) = 1, then \F-k\ = (1)/(n)((n)(m1,m2,....,mr)) for eac h 0 less than or equal to k < n. Finally, some applications of these proper ties concerning the q-multinomial coefficient are found. (C) 2001 Academic Press.