Standard Rothe Tableaux

TL;DR

This paper introduces standard Rothe tableaux for permutations, compares their count to balanced Rothe tableaux, and characterizes when counts are equal based on pattern avoidance, extending classical results for dominant permutations.

Contribution

It defines standard Rothe tableaux, establishes their enumeration relationship with balanced Rothe tableaux, and characterizes equality cases via pattern avoidance.

Findings

Number of standard Rothe tableaux ≤ number of balanced Rothe tableaux.

Equality holds iff permutation avoids four specific patterns.

For dominant permutations, the result reduces to Edelman-Greene's classical case.

Abstract

Edelman and Greene constructed a bijection between the set of standard Young tableaux and the set of balanced Young tableaux of the same shape. Fomin, Greene, Reiner and Shimozono introduced the notion of balanced Rothe tableaux of a permutation $w$, and established a bijection between the set of balanced Rothe tableaux of $w$ and the set of reduced words of $w$. In this paper, we introduce the notion of standard Rothe tableaux of $w$, which are tableaux obtained by labelling the cells of the Rothe diagram of $w$ such that each row and each column is increasing. We show that the number of standard Rothe tableaux of $w$ is smaller than or equal to the number of balanced Rothe tableaux of $w$, with equality if and only if $w$ avoids the four patterns 2413, 2431, 3142 and 4132. When $w$ is a dominant permutation, i.e., 132-avoiding, the Rothe diagram of $w$ is a Young diagram, so this reduces to the result of Edelman and Greene.