scholarly journals Refinements and Symmetries of the Morris identity for volumes of flow polytopes

2021 ◽  
Vol 359 (7) ◽  
pp. 823-851
Author(s):  
Alejandro H. Morales ◽  
William Shi
Keyword(s):  
Flow Polytopes

scholarly journals Flow polytopes and the graph of reflexive polytopes

2009 ◽  
Vol 309 (16) ◽  
pp. 4992-4999 ◽  
Author(s):  
Klaus Altmann ◽  
Benjamin Nill ◽  
Sabine Schwentner ◽  
Izolda Wiercinska
Keyword(s):  
Flow Polytopes ◽  
Reflexive Polytopes

scholarly journals A combinatorial model for computing volumes of flow polytopes

2019 ◽  
Vol 372 (5) ◽  
pp. 3369-3404 ◽  
Author(s):  
Carolina Benedetti ◽  
Rafael S. González D’León ◽  
Christopher R. H. Hanusa ◽  
Pamela E. Harris ◽  
Apoorva Khare ◽  
...  
Keyword(s):  
Flow Polytopes ◽  
Combinatorial Model

scholarly journals Flow polytopes with Catalan volumes

2017 ◽  
Vol 355 (3) ◽  
pp. 248-259 ◽  
Author(s):  
Sylvie Corteel ◽  
Jang Soo Kim ◽  
Karola Mészáros
Keyword(s):  
Flow Polytopes

scholarly journals Flow Polytopes and the Space of Diagonal Harmonics

2019 ◽  
Vol 71 (6) ◽  
pp. 1495-1521
Author(s):  
Ricky Ini Liu ◽  
Alejandro H. Morales ◽  
Karola Mészáros
Keyword(s):  
Complete Graph ◽  
Hilbert Series ◽  
Threshold Graphs ◽  
Flow Polytopes ◽  
Diagonal Harmonics ◽  
Flow Polytope

AbstractA 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 vector$(-n,1,\ldots ,1)$. We study the$(q,t)$-Ehrhart functions of flow polytopes of threshold graphs with arbitrary netflow vectors. Our results generalize previously known specializations of the mentioned bigraded Hilbert series at$t=1$,$0$, and$q^{-1}$. As a corollary to our results, we obtain a proof of a conjecture of Armstrong, Garsia, Haglund, Rhoades, and Sagan about the$(q,q^{-1})$-Ehrhart function of the flow polytope of a complete graph with an arbitrary netflow vector.

scholarly journals Flow Polytopes of Partitions

10.37236/8114 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Karola Mészáros ◽  
Connor Simpson ◽  
Zoe Wellner
Keyword(s):  
Recent Progress ◽  
Combinatorial Type ◽  
Flow Polytopes ◽  
Work Breaks ◽  
Product Of Simplices ◽  
Product Formulas

Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes $F_{(\lambda, {\bf a})}$ for each partition shape $\lambda$ and netflow vector ${\bf a}\in Z^n_{> 0}$. In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed family $F_{(\lambda, {\bf a})}$ is the same. When $\lambda$ is a staircase shape and ${\bf a}$ is the all ones vector the latter results specializes to a theorem of the first author with Morales and Rhoades, which shows that the combinatorial type of the Tesler polytope is a product of simplices.

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

2019 ◽  
Vol 62 (1) ◽  
pp. 128-163 ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales ◽  
Jessica Striker
Keyword(s):  
Flow Polytopes ◽  
Alternating Sign Matrix ◽  
Sign Matrix

scholarly journals Volumes and Ehrhart polynomials of flow polytopes

2019 ◽  
Vol 293 (3-4) ◽  
pp. 1369-1401 ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales
Keyword(s):  
Ehrhart Polynomials ◽  
Flow Polytopes

scholarly journals Flow polytopes and the Kostant partition function

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Karola Mészáros ◽  
Alejandro H. Morales
Keyword(s):  
Partition Function ◽  
Catalan Number ◽  
Signed Graphs ◽  
Volume Formula ◽  
Flow Polytopes ◽  
International Audience ◽  
Kostant Partition Function ◽  
Mise En Pratique ◽  
The Relationship ◽  
Special Case

International audience 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 combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As an application of our results we study a distinguished family of flow polytopes: the Chan-Robbins-Yuen polytopes. Inspired by their beautiful volume formula $\prod_{k=0}^{n-2} Cat(k)$ for the type $A_n$ case, where $Cat(k)$ is the $k^{th}$ Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures about their volumes. Nous établissons la relation entre les volumes de polytopes de flux associés aux graphes signés et la fonction de partition de Kostant. Le cas particulier de cette relation où les graphes ne sont pas signés a été étudié en détail par Baldoni et Vergne en utilisant des techniques de résidus. Contrairement à leur approche, nous apportons des preuves combinatoires inspirées par l'analyse de Postnikov et Stanley sur les polytopes de flux. Comme mise en pratique des résultats, nous étudions une famille distinguée de polytopes de flux: les polytopes Chan-Robbins-Yuen. Inspirés par leur belle formule du volume $\prod_{k=0}^{n-2} Cat(k)$ pour le cas de type $A_n$ (où $Cat(k)$ est le $k$-ème nombres de Catalan), nous présentons les polytopes Chan-Robbins-Yuen des types $C_{n +1}$ et $D_{n +1}$ accompagnés de conjectures intéressantes sur leurs volumes.

Sign in / Sign up

Export Citation Format

Share Document