Lattice Point Generating Functions and Symmetric Cones

TL;DR

This paper generalizes a known identity for generating functions of symmetric compositions to convex polyhedral cones invariant under finite reflection groups, providing new formulas and applications for types B and D.

Contribution

It extends existing identities to a broader class of cones and derives new formulas for types B and D, with applications in permutation statistics and lecture hall partitions.

Findings

General formulas for multivariate generating functions of symmetric cones

Explicit results for types B and D reflection groups

Applications to permutation statistics and lecture hall partitions

Abstract

We show that a recent identity of Beck-Gessel-Lee-Savage on the generating function of symmetrically contrained 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 the specific cases of a symmetry group of type A (previously known) and types B and D (new). We obtain several applications of the special cases in type B, including identities involving permutation statistics and lecture hall partitions.