Major Index for 01-Fillings of Moon Polyominoes

TL;DR

This paper introduces a major index statistic for 01-fillings of moon polyominoes, generalizing classical permutation statistics and establishing its equidistribution with north-east chains through algebraic and bijective proofs.

Contribution

It defines a new major index for moon polyomino fillings and proves its distributional equivalence to north-east chains, extending classical permutation results.

Findings

Major index matches the distribution of north-east chains.

The results generalize classical permutation statistics.

Two proofs: algebraic and bijective, are provided.

Abstract

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s; A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations.