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

Hugh Denoncourt, Dana C Ernst, Dustin Story


Original proposers of the open problem: Donald E. Knuth
The year when the open problem was proposed: 1992
Sponsor of the submission: Richard M. Green - University of Colorado Boulder
AMS Subject classification: 05
Status of the problem: Open

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 $S_n$?

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