An extension of MacMahon's equidistribution theorem to ordered multiset partitions

作者：Wilson Andrew Timothy^*

来源：ELECTRONIC JOURNAL OF COMBINATORICS, 2016, 23(1): P1.5.

DOI：10.37236/5485

• inversion number
• ordered multiset partitions
• permutation statistics
• insertion method
• major index
• Macdonald polynomials

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.

• 出版日期2016-1-11