On the number of commutation classes of the longest element in the symmetric group

Abstract

Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in $S_n$?

Figures (4)

• Figure 1: Reduced expressions and the corresponding commutation classes for the longest element in $S_4$.
• Figure 2: Heaps for the longest element in $S_4$.
• Figure 3: Minimal ladder lotteries corresponding to the primitive sorting networks on 4 elements.
• Figure 4: Rhombic tilings of a regular octagon.