Combinatorics of Tableau Inversions

TL;DR

This paper explores the combinatorial properties of tableau inversions, providing formulas for counting inverted Young tableaux of various shapes, establishing bijections between different shapes, and sharing computational tools for analysis.

Contribution

It introduces new combinatorial formulas and bijections for counting and relating inverted Young tableaux of different shapes, expanding understanding beyond previous algebraic approaches.

Findings

Formulas for enumerating $i$-inverted Young tableaux for various shapes

Bijections between $i$-inverted tableaux of different shapes

Development of a computer program to calculate tableaux inversions

Abstract

A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with a precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau inversions from a completely combinatorial perspective. We develop formulas enumerating the number of $i$-inverted Young tableaux for a variety of tableaux shapes, not restricting ourselves to inverted tableaux that standardize a specific standard Young tableau, and construct bijections between $i$-inverted Young tableaux of a certain shape with $j$-inverted Young tableaux of different shapes. Finally, we share some the results of a computer program developed to calculate tableaux inversions.