An extension of MacMahon's equidistribution theorem to ordered multiset partitions
The electronic journal of combinatorics, Tome 23 (2016) no. 1
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. A corrigendum was added 17 September 2019.
DOI :
10.37236/5485
Classification :
05A19, 05A05, 05A18, 05E05
Mots-clés : inversion number, major index, permutation statistics, insertion method, ordered multiset partitions, Macdonald polynomials
Mots-clés : inversion number, major index, permutation statistics, insertion method, ordered multiset partitions, Macdonald polynomials
Affiliations des auteurs :
Andrew Timothy Wilson 1
@article{10_37236_5485,
author = {Andrew Timothy Wilson},
title = {An extension of {MacMahon's} equidistribution theorem to ordered multiset partitions},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {1},
doi = {10.37236/5485},
zbl = {1329.05030},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5485/}
}
Andrew Timothy Wilson. An extension of MacMahon's equidistribution theorem to ordered multiset partitions. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5485
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