Lattice point generating functions and symmetric cones

Matthias Beck, Thomas Bliem, Benjamin Braun, Carla D. Savage

Research output: Contribution to journalArticlepeer-review

Abstract

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically constrained compositions of integers generalizes naturally to a family of convex polyhedral cones that are invariant under the action of a finite reflection group. We obtain general expressions for the multivariate generating functions of such cones, and work out their general form more specifically for all symmetry groups of type A (previously known) and types B and D (new). We obtain several applications of these expressions in type B, including identities involving permutation statistics and lecture hall partitions.

Original languageEnglish
Pages (from-to)543-566
Number of pages24
JournalJournal of Algebraic Combinatorics
Volume38
Issue number3
DOIs
StatePublished - Nov 2013

Bibliographical note

Funding Information:
Acknowledgements We thank the anonymous referees for their thoughtful comments. M.B. is partially supported by the NSF (DMS-0810105 & DMS-1162638). T.B. has been supported by the Deutsche Forschungsgemeinschaft (SPP 1388). B.B. is partially supported by the NSF (DMS-0758321). We are grateful to the American Institute of Mathematics for supporting our SQuaRE working group “Polyhedral Geometry and Partition Theory” where our collaboration on this work began.

Keywords

  • Coxeter group
  • Finite reflection group
  • Lattice point generating function
  • Lecture hall partition
  • Permutation statistics
  • Polyhedral cone
  • Symmetrically constrained composition

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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