# Flow polytopes with Catalan volumes

@article{Corteel2016FlowPW, title={Flow polytopes with Catalan volumes}, author={Sylvie Corteel and Jang Soo Kim and Karola M'esz'aros}, journal={Comptes Rendus Mathematique}, year={2016}, volume={355}, pages={248-259} }

Abstract The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector ( 1 , 0 , … , 0 , − 1 ) . The normalized volume of the Chan–Robbins–Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector ( 1 , 1 , 0 , … , 0 , − 2 ) . We show that the volume of this polytope… Expand

#### 13 Citations

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… Expand

A combinatorial model for computing volumes of flow polytopes

- Mathematics
- Transactions 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… Expand

Volumes of Generalized Chan–Robbins–Yuen Polytopes

- Computer Science, Mathematics
- Discret. Comput. Geom.
- 2021

The normalized volume of the Chan–Robbins–Yuen polytope is the product of consecutive Catalan numbers and Zeilberger proved one of these conjectures and proofs of both conjectures are presented. Expand

A Fuss-Catalan variation of the caracol flow polytope

- Mathematics
- 2019

Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of… Expand

Volumes of Flow Polytopes Related to Caracol Graphs

- Mathematics, Computer Science
- Electron. J. Comb.
- 2020

Benedetti et al.'s conjecture for the Ehrhart-like polynomial of what they call a caracol graph is proved using constant term identities, labeled Dyck paths, and a cyclic lemma. Expand

Volumes and Ehrhart polynomials of flow polytopes

- Mathematics
- Mathematische 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… Expand

The volume of the caracol polytope

- 2019

We give a combinatorial interpretation of the Lidskii formula for flow polytopes and use it to compute volumes via the enumeration of new families of combinatorial objects which are generalizations… Expand

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

- Mathematics
- 2021

We study polytopes defined by inequalities of the form ∑ i∈I zi ≤ 1 for I ⊆ [d] and nonnegative zi where the inequalities can be reordered into a matrix inequality involving a column-convex {0,… Expand

Extensions of partial cyclic orders and consecutive coordinate polytopes

- Mathematics
- Annales Henri Lebesgue
- 2020

We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates, extending a construction by Stanley. We show that the… Expand

On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

- Computer Science
- Electron. J. Comb.
- 2020

The volume of a special kind of flow polytope is considered and it is shown that its volume satisfies a certain system of differential equations, and conversely, the solution of the systemof differential equations is unique up to a constant multiple. Expand

#### References

SHOWING 1-10 OF 14 REFERENCES

Flow polytopes of signed graphs and the Kostant partition function

- Mathematics
- 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… Expand

On the Volume of a Certain Polytope

- Mathematics, Computer Science
- Exp. 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. Expand

Quivers, cones and polytopes

- Mathematics
- 2003

Abstract Let Q be a quiver without oriented cycles. We consider the polytope of flows Δ( θ ) in Q with input θ . These polytopes are closely related to the combinatorial structure of the quiver, in… Expand

On the Volume of the Polytope of Doubly Stochastic Matrices

- Mathematics, Computer Science
- Exp. 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”. Expand

Flow Polytopes and the Space of Diagonal Harmonics

- Mathematics
- Canadian Journal of Mathematics
- 2019

Abstract A result of Haglund implies that the $(q,t)$ -bigraded Hilbert series of the space of diagonal harmonics is a $(q,t)$ -Ehrhart function of the flow polytope of a complete graph with netflow… Expand

Pipe Dream Complexes and Triangulations of Root Polytopes Belong Together

- Computer Science, Mathematics
- SIAM 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. Expand

Product formulas for volumes of flow polytopes

- Mathematics
- 2011

Intrigued by the product formula prod_{i=1}^{n-2} C_i for the volume of the Chan-Robbins-Yuen polytope CRY_n, where C_i is the ith Catalan number, we construct a family of polytopes P_{m,n}, whose… Expand

Subword complexes via triangulations of root polytopes

- Mathematics
- 2015

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is… Expand

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… Expand

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… Expand