Restricted Birkhoff polytopes and Ehrhart period collapse

TL;DR

This paper demonstrates that certain restricted Birkhoff polytopes have Ehrhart polynomials, not just quasi-polynomials, due to a bijection with Gelfand-Tsetlin polytopes that preserves lattice points.

Contribution

It introduces a new class of polytopes with Ehrhart polynomial properties and establishes a bijection related to the RSK correspondence that explains this phenomenon.

Findings

Restricted Birkhoff polytopes have Ehrhart polynomials.

A piecewise-linear bijection to Gelfand-Tsetlin polytopes is constructed.

The bijection explains Ehrhart period collapse in these polytopes.

Abstract

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand-Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson-Schensted-Knuth correspondence.