Volumes of generalized Chan-Robbins-Yuen polytopes

Abstract

The normalized volume of the Chan-Robbins-Yuen polytope ($CR{Y}_n$) is the product of consecutive Catalan numbers. The polytope $CR{Y}_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its volume formula. In their quest to understand $CR{Y}_n$ 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.

15 pages, 3 figures

References in corpus (3)

• Flow polytopes of signed graphs and the Kostant partition function
• The polytope of Tesler matrices
• Sketch of a Proof of an Intriguing Conjecture of Karola Meszaros and Alejandro Morales Regarding the Volume of the $D_n$ Analog of the Chan-Robbins-Yuen Polytope (Or: The Morris-Selberg Constant Term Identity Strikes Again!)