Column convex matrices, $G$-cyclic orders, and flow polytopes

TL;DR

This paper establishes a connection between certain polytopes defined by inequalities, flow polytopes of special graphs, and combinatorial numbers like Euler numbers, providing new formulas and conjectures for their properties.

Contribution

It introduces an integral equivalence between column convex polytopes and flow polytopes of spinal graphs, and explores their combinatorial and geometric properties.

Findings

Flow polytope volumes equal the number of extensions of partial cyclic orders.

Refinements of $k$-Euler numbers are realized as Kostant partition function values.

A formula for the $h^*$-polynomial of flow polytopes is provided, with conjectures for bounds.

Abstract

We study polytopes defined by inequalities of the form $\sum_{i\in I} z_{i}\leq 1$ for $I\subseteq [d]$ and nonnegative $z_i$ where the inequalities can be reordered into a matrix inequality involving a column-convex $\{0,1\}$-matrix. These generalize polytopes studied by Stanley, and the consecutive coordinate polytopes of Ayyer, Josuat-Verg\`es, and Ramassamy. We prove an integral equivalence between these polytopes and flow polytopes of directed acyclic graphs $G$ with a Hamiltonian path, which we call spinal graphs. We show that the volume of these flow polytopes is the number of extensions of a set of partial cyclic orders defined by the graph $G$. As a special case we recover results on volumes of consecutive coordinate polytopes. We study the combinatorics of $k$-Euler numbers, which are generalizations of the classical Euler numbers, and which arise as volumes of flow polytopes of a special family of spinal graphs. We show that their refinements, Ramassamy's $k$-Entringer numbers, can be realized as values of a Kostant partition function, satisfy a family of generalized boustrophedon recurrences, and are log concave along root directions. Finally, via our main integral equivalence and the known formula for the $h^*$-polynomial of consecutive coordinate polytopes, we give a combinatorial formula for the $h^*$-polynomial of flow polytopes of non-nested spinal graphs. For spinal graphs in general, we present a conjecture on upper and lower bounds for their $h^*$-polynomial.