# Flow polytopes of signed graphs and the Kostant partition function

@article{Mszros2012FlowPO, title={Flow polytopes of signed graphs and the Kostant partition function}, author={Karola M{\'e}sz{\'a}ros and Alejandro H. Morales}, journal={arXiv: Combinatorics}, year={2012} }

We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins…

## 29 Citations

Volumes and Ehrhart polynomials of flow polytopes

- MathematicsMathematische Zeitschrift
- 2019

The Lidskii formula for the type $$A_n$$An root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant…

Flow Polytopes of Partitions

- Mathematics, Computer ScienceElectron. J. Comb.
- 2019

This work defines a family of closely related flow polytopes and proves that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes.

REFINEMENTS OF PRODUCT FORMULAS FOR VOLUMES OF FLOW POLYTOPES

- 2020

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan–Robbins–Yuen (CRY) polytope is a flow polytope…

A combinatorial model for computing volumes of flow polytopes

- MathematicsTransactions of the American Mathematical Society
- 2019

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…

On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2019

It is proved by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes.

Counting Integer Points of Flow Polytopes

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2021

The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes, and to reveal the geometry of these formulas.

h-Polynomials via Reduced Forms

- Mathematics, Computer ScienceElectron. J. Comb.
- 2015

It is proved that reduced forms in the subdivision algebra are generalizations of $h-polynomials of the triangulations of flow polytopes, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced form in the noncommutative quasi-classical Yang-Baxter algebra.

Pipe Dream Complexes and Triangulations of Root Polytopes Belong Together

- Computer Science, MathematicsSIAM J. Discret. Math.
- 2016

The Grothendieck polynomials are connected to reduced forms in subdivision algebras and root (and flow) poly topes, explaining that these families of polytopes possess the same subdivision algebra.

Toric matrix Schubert varieties and root polytopes (extended abstract)

- Mathematics
- 2016

Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix…

## References

SHOWING 1-10 OF 16 REFERENCES

Partial fraction decompositions and an algorithm for computing the vector partition function

- Mathematics
- 2009

This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer…

On the Volume of a Certain Polytope

- Mathematics, Computer ScienceExp. Math.
- 2000

The convex hull Pn of Tn, a polytope of dimension (n 2), is studied, providing evidence for several conjectures involving Pn, including Conjecture 1: Let Vn denote the minimum volume of a simplex with vertices in the affine lattice spanned by Tn.

Kostant Partitions Functions and Flow Polytopes

- Mathematics
- 2008

This paper discusses volumes and Ehrhart polynomials in the context of flow polytopes. The general approach that studies these functions via rational functions with poles on arrangement of…

On the Volume of the Polytope of Doubly Stochastic Matrices

- Mathematics, Computer ScienceExp. Math.
- 1999

This work studies the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one), and describes two methods for the enumeration of “magic squares”.

Product formulas for volumes of flow polytopes

- Mathematics
- 2014

We outline the construction of a family of polytopes \(\mathcal{P}_{m,n}\), indexed by \(m \in \mathbb{Z}_{\geq 0}\) and \(n \in \mathbb{Z}_{\geq 2}\), whose volumes are given by the product
…

Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra

- Mathematics
- 2007

Preface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville…

Root polytopes, triangulations, and the subdivision algebra. I

- Mathematics
- 2009

The type A n root polytope P(A + n ) is the convex hull in R n+1 of the origin and the points e i — e j for 1 ≤ i < j ≤ n + 1. Given a tree T on the vertex set [n + 1], the associated root polytope…

Residues formulae for volumes and Ehrhart polynomials of convex polytopes.

- Mathematics
- 2001

In these notes, we explain residue formulae for volumes of convex polytopes, and for Ehrahrt polynomials based on the notion of total residue. We apply this method to the computation of the volume of…

Morris identities and the total residue for a system of type A r

- Physics
- 2004

The purpose of this paper is to find explicit formulae for the total residue of some interesting rational functions with poles on hyperplanes determined by roots of type A r = {(e i −e j )|1 ≤ i, j ≤…

Proof of a Conjecture of Chan, Robbins, and Yuen

- Mathematics
- 1998

Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain $n(n-1)/2$-dimensional…