Statistics on Multisets

TL;DR

This paper provides a new proof linking q-analogues of multinomial coefficients to permutation inversion counts in multisets, using vector space chain enumeration over finite fields.

Contribution

It introduces a novel proof connecting q-multinomial coefficients with permutation inversion enumeration via vector space chains.

Findings

Q-analogues of multinomial coefficients count permutations by inversions.

The proof uses chains of subspaces in finite vector spaces.

The paper investigates permutation counts with fixed inversion numbers.

Abstract

We offer a new proof that a certain q-analogue of multinomial coeffi- cients furnishes a q-counting of the set of permutations of an associated multiset of positive integers, according to the number of inversions in such arrangements. Our proof uses the fact that such q-multinomial coefficients enumerate certain classes of chains of subspaces of a fnite dimensional vector space over a fnite field of cardinality q. Additionally, we investigate the function that counts the number of permutations of a multiset having a fixed number of inversions.