Counting Lattice Points in Generalized Permutohedra From A to B

TL;DR

This paper presents a new formula for counting lattice points in type B generalized permutohedra, extending existing methods and linking to Ehrhart polynomials and volume calculations.

Contribution

It introduces a concise lattice point counting formula for type B generalized permutohedra using Postnikov's G-draconian sequences, expanding the combinatorial understanding.

Findings

Derived a new lattice point count formula for type B permutohedra

Connected lattice point counts to Ehrhart polynomials and volume

Provided an alternative to recent formulas by Eur et al.

Abstract

We derive a formula for the number of lattice points in type B generalized permutohedra, providing a concise alternative to the formula obtained recently by Eur, Fink, Larson, and Spink as a result from a study of delta-matroids. Our approach builds upon the existing framework and techniques introduced by Postnikov in his work on type A generalized permutohedra, a family of polytopes interconnected with many mathematical concepts such as matroids and Weyl groups. In particular, we express the number of lattice points in type B generalized permutohedra in terms of Postnikov's notion of G-draconian sequences, from which their Ehrhart polynomials and volume formula follow as consequences.