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Standard Rothe Tableaux

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.