  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.
