Let be the multiset that has copies of , . denotes the set of permutations of for all distinct orderings of the integers. The number of elements in is the multinomial coefficient (§26.4) . Additional information can be found in Andrews (1976, pp. 39–45).
The definitions of inversion number and major index can be extended to permutations of a multiset such as . Thus , and
The -multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
and again with we have