Lattice point generating functions and symmetric cones
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.
KeywordsLattice point generating function Polyhedral cone Finite reflection group Coxeter group Symmetrically constrained composition Permutation statistics Lecture hall partition
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.
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