On a duality in Coxeter groups

  • Authors:

  • Mario Marietti

  • Affiliations:

  • Dipartimento di Matematica, "Sapienza" Universití di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2008

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Abstract

In [R.P. Stanley, The descent set and connectivity set of a permutation, J. Integer Seq. 8 (3) (2005) Article 05.3.8] Stanley gives certain enumerative identities revealing a duality between descent sets and connectivity sets of the symmetric group. In this paper we generalize these identities to all Coxeter groups. The proofs are obtained by giving these identities an algebraic explanation in terms of parabolic subgroups, coset representatives, and Poincare series, and by a formal argument in terms of inclusion-exclusion-like matrices.