Combinatorics of Tableau Inversions

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 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 tableau 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