The number of inversions of permutations with fixed shape

TL;DR

This paper investigates the distribution of inversion counts among permutations mapped to a fixed shape via the Robinson-Schensted correspondence, providing new results and conjectures especially for permutations with higher inversion numbers.

Contribution

It offers the first results for permutations with more inversions than the minimum, including explicit conjectures and a proof for shapes with two columns.

Findings

Characterized permutations with shape λ and higher inversions.

Proposed conjectures for permutations with inversions less than the smallest column length.

Proved results for shapes with two columns.

Abstract

The Robinson-Schensted correspondence can be viewed as a map from permutations to partitions. In this work, we study the number of inversions of permutations corresponding to a fixed partition $\lambda$ under this map. Hohlweg characterized permutations having shape $\lambda$ with the minimum number of inversions. Here, we give the first results in this direction for higher numbers of inversions. We give explicit conjectures for both the structure and the number of permutations associated to $\lambda$ where the extra number of inversions is less than the length of the smallest column of $\lambda$. We prove the result when $\lambda$ has two columns.