On distribution of the depth index on perfect matchings

TL;DR

This paper explores the distribution of the depth index on perfect matchings, providing combinatorial insights and establishing its equidistribution with the Bruhat order's rank function.

Contribution

It offers a new combinatorial description of the depth index and proves its equidistribution with the Bruhat order rank function for perfect matchings.

Findings

Depth index is equidistributed with the Bruhat order rank function.

Provided a combinatorial description of the depth index.

Calculated the generating polynomial for the depth index.

Abstract

We study the restriction of depth index statistic on the set of perfect matchings. In particular, we provide additional combinatorial description of the statistic for perfect matchings and calculate the generating polynomial. The main result of the present short paper is that the depth index on perfect matchings is equidistributed with the rank function of the Bruhat order.