Counting Lattice Points in Generalized Permutohedra From A to B
Warut Thawinrak
TL;DR
This work extends Postnikov’s type A framework for counting lattice points to type B generalized permutohedra by decomposing B-polytopes into octants that each realize as type A permutohedra. It introduces and leverages G-draconian sequences to enumerate lattice points, establishing bijections with zonotopal fine mixed cells and yielding explicit formulas for lattice-point counts, Ehrhart polynomials, and volumes. The resulting approach provides a concise alternative to the recent delta-matroid-based formula and shows that many type B properties can be studied through the lens of type A techniques. The paper also outlines several promising directions for extending these methods, including face combinatorics, polynomial invariants, and positivity questions in the type B setting.
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.