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.
@article{10_37236_8114,
author = {Karola M\'esz\'aros and Connor Simpson and Zoe Wellner},
title = {Flow polytopes of partitions},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {1},
doi = {10.37236/8114},
zbl = {1409.52013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8114/}
}
TY - JOUR
AU - Karola Mészáros
AU - Connor Simpson
AU - Zoe Wellner
TI - Flow polytopes of partitions
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8114/
DO - 10.37236/8114
ID - 10_37236_8114
ER -
%0 Journal Article
%A Karola Mészáros
%A Connor Simpson
%A Zoe Wellner
%T Flow polytopes of partitions
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8114/
%R 10.37236/8114
%F 10_37236_8114
Karola Mészáros; Connor Simpson; Zoe Wellner. Flow polytopes of partitions. The electronic journal of combinatorics, Tome 26 (2019) no. 1. doi: 10.37236/8114
