Simultaneous maj statistics (Q1290738)

It is well known that the major index statistic on words has the Mahonian distribution. This paper considers a trinomial variation on this theme: For a word \(w\) formed from a three-letter alphabet \(\{0,1,2\}\), assume the ordering \(0> 1> 2\) and set \[ \text{maj}_{012}(w)= \sum_{i: w_i> w_{i+1}}i. \] Define \(\text{maj}_{120}\) and \(\text{maj}_{201}\) analogously and consider the trinomial generating function \[ \sum_w x^{\text{maj}_{120}(w)} y^{\text{maj}_{201}(w)} z^{\text{maj}_{012}(w)}. \] The authors use induction to verify a closed form expression for this generating function as a sum of \(xyz\)-trinomial coefficients. The theorem described above generalizes to a seven-variable version which in turn specializes to 48 different Mahonian statistics. The ultimate theorem generalizes to words on \(N\) letters yielding \(N!2^N\) Mahonian statistics as specializations. The paper concludes with an interpretation of these generating functions for \(N\)-tuples of integer partitions with no (pairwise) common parts.