Inversions of Semistandard Young Tableaux

TL;DR

This paper introduces a new generalization of semistandard Young tableaux through the concept of tableau inversions, providing formulas and enumeration methods for these objects with various shapes and contents.

Contribution

It extends the notion of tableau inversions to row-standard tableaux with repeated entries, offering a closed formula for maximum inversion pairs and invariance under content permutation.

Findings

Derived a closed formula for maximum inversion pairs in specific shapes

Proved invariance of inversion counts under content permutation

Enumerated inverted Young tableaux for various shapes and contents

Abstract

A tableau inversion is a pair of entries from the same column of a row-standard tableau that lack the relative ordering necessary to make the tableau column-standard. An $i$-inverted Young tableau is a row-standard tableau with precisely $i$ inversion pairs, and may be interpreted as a generalization of (column-standard) Young tableau. Inverted Young tableau that lack repeated entries were introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, and were later developed as combinatorial objects in their own right by Beagley and Drube. This paper generalizes earlier notions of tableau inversions to row-standard tableaux with repeated entries, yielding an interesting new generalization of semistandard (as opposed to merely standard) Young tableaux. We develop a closed formula for the maximum numbers of inversion pairs for a row-standard tableau with a specific shape and content, and show that the number of $i$-inverted tableaux of a given shape is invariant under permutation of content. We then enumerate $i$-inverted Young tableaux for a variety of shapes and contents, and generalize an earlier result that places $1$-inverted Young tableaux of a general shape in bijection with $0$-inverted Young tableaux of a variety of related shapes.