Number of vertices in Gelfand-Zetlin polytopes

TL;DR

This paper investigates the enumeration of vertices in Gelfand-Zetlin polytopes by deriving a differential equation for their generating function and providing explicit formulas for specific classes.

Contribution

It introduces a differential equation approach for counting vertices and offers explicit formulas for certain classes of Gelfand-Zetlin polytopes, advancing combinatorial enumeration methods.

Findings

Derived a partial differential equation for the generating function of vertex counts.

Provided explicit formulas for the number of vertices in particular classes of Gelfand-Zetlin polytopes.

Enhanced understanding of the combinatorial structure of Gelfand-Zetlin polytopes.

Abstract

We discuss the problem of counting vertices in Gelfand-Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of Gelfand-Zetlin polytopes, the number of vertices can be given by explicit formulas.