# On the number of faces of Gelfand-Zetlin polytopes

## TL;DR

This paper derives a recurrence relation for the f-polynomial of Gelfand-Zetlin polytopes using geometric analysis, leading to explicit formulas for their f- and h-polynomials in certain families.

## Contribution

It introduces a new recurrence relation for Gelfand-Zetlin polytopes' f-polynomials based on geometric projections, enabling explicit formula derivation.

## Key findings

- Derived recurrence relation for f-polynomial
- Explicit formulas for f- and h-polynomials in specific families
- Enhanced understanding of Gelfand-Zetlin polytope combinatorics

## Abstract

We obtain a recurrence relation for the f-polynomial of Gelfand-Zetlin polytopes by analyzing geometric properties of a linear projection of the Gelfand-Zetlin polytope onto a cube. We apply this recurrence relation to find explicit formulas for the f-polynomials and h-polynomials of Gelfand-Zetlin polytopes that belong to several simplest one-parameter families.