Combinatorics of tableau inversions

Jonathan E. Beagley¹ ; Paul Drube¹

¹Valparaiso University

The electronic journal of combinatorics, Tome 22 (2015) no. 2

Résumé

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.

Détail

DOI : 10.37236/4932

Classification : 05A19, 05A05
Mots-clés : Young tableaux, inversions of Young tableaux