Combinatorics of Tableaux Inversions

Jonathan E. Beagley, Paul Drube

Published 2014-12-19Version 1

A tableaux inversion is a pair of entries in row-standard tableaux $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableaux is a row-standard tableaux along with a precisely $i$-inversion pairs. Tableaux 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 tableaux corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableaux 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 tableaux, 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.