The Enumeration of Fully Commutative Elements of Coxeter Groups

Authors:
John R. Stembridge
Affiliations:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1109. E-mail: jrs@math.lsa.umich.edu
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
1998

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Abstract

A Coxeter group element w is fully commutative if anyreduced expression for w can be obtained from any other via theinterchange of commuting generators. For example, in the symmetric group ofdegree n, the number of fully commutative elements is thenth Catalan number. The Coxeter groups with finitely many fullycommutative elements can be arranged into seven infinite familiesA_n, B_n, D_n, E_n, F_n, H_n and I_2(m). For each family,we provide explicit generating functions for the number of fully commutativeelements and the number of fully commutative involutions; in each case, thegenerating function is algebraic.