  The combinatorics of reduced words and commutation classes plays an important
role in geometric representation theory. A string polytope is a lattice
polytope associated to each reduced word of the longest element $w_0$ in the
symmetric group which encodes the character of a certain irreducible
representation of a Lie group of type $A$. In this paper, we provide a
recursive formula for the number of reduced words of $w_0$ such that the
corresponding string polytopes are combinatorially equivalent to a
Gelfand-Cetlin polytope. The recursive formula involves the number of standard
Young tableaux of shifted shape. We also show that each commutation class is
completely determined by a list of quantities called indices.
