A COMBINATORIAL MODEL FOR COMPUTING VOLUMES OF FLOW POLYTOPES

Benedetti, Carolina and Gonzalez D Leon, Rafael S and Hanusa, Christopher R H and Harris, Pamela E and Khare, Apoorva and Morales, Alejandro H and Yip, Martha (2019) A COMBINATORIAL MODEL FOR COMPUTING VOLUMES OF FLOW POLYTOPES. In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 372 (5). pp. 3369-3404.

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Abstract

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne's generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.

Item Type:	Journal Article
Publication:	TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Publisher:	AMER MATHEMATICAL SOC
Additional Information:	Copyright of this article belongs to AMER MATHEMATICAL SOC
Keywords:	Flow polytope; parking function; Lidskii formula; Kostant partition function; caracol graph; Chan-Robbins-Yuen polytope; Tesler polytope; Pitman-Stanley polytope; zigzag graph; line-dot diagram; gravity diagram; unified diagram; log-concave; Catalan numbers; parking triangle; binomial transform; Dyck path; multi-labeled Dyck path; Ehrhart polynomial
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	18 Nov 2019 11:59
Last Modified:	18 Nov 2019 11:59
URI:	http://eprints.iisc.ac.in/id/eprint/63441

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