# Volumes of generalized Chan-Robbins-Yuen polytopes

## TL;DR

This paper proves two conjectures regarding the volumes of generalized Chan-Robbins-Yuen polytopes, showing they are products of Catalan numbers and powers of two, thus advancing understanding of these combinatorial structures.

## Contribution

The paper provides the first proofs of two conjectures about the volumes of generalized CRY polytopes, confirming their formulas involving Catalan numbers and powers of two.

## Key findings

- Both conjectures about the volumes are proven.
- Volumes are expressed as products of Catalan numbers and powers of two.
- Results deepen understanding of the combinatorial properties of these polytopes.

## Abstract

The normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CRY_n$ better, the third author and Morales introduced two natural generalizations of it and conjectured that their volumes are certain powers of $2$ multiplied by a product of consecutive Catalan numbers. Zeilberger proved one of these conjectures. In this paper we present proofs of both conjectures.