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.
|Number of pages
|Journal of Algebraic Combinatorics
|Published - Nov 2013
Bibliographical noteFunding 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.
- 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