Abstract
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.
Highlights
The Catalan numbers, Cn = 1 n+1 n 2, n∈Z0, are well known for counting a plethora of combinatorial objects; see [6, Ex. 6.19] for hundreds of interpretations
If an integer polytope has volume divisible by a product of consecutive Catalan numbers, one would hope for a combinatorial explanation of such a phenomenon
The latter sentiment ran into obstacles with several flow polytopes, namely the Chan-Robbins-Yuen polytope [2], its type C and D generalizations [4], as well as the Tesler polytope [5]
Summary
0, are well known for counting a plethora of combinatorial objects; see [6, Ex. 6.19] for hundreds of interpretations. If an integer polytope has volume divisible by a product of consecutive Catalan numbers, one would hope for a combinatorial explanation of such a phenomenon. The latter sentiment ran into obstacles with several flow polytopes, namely the (type A) Chan-Robbins-Yuen polytope [2], its type C and D generalizations [4], as well as the Tesler polytope [5]. The limiting polytope of F(λ,a) is integrally equivalent to a product of scaled simplices a1∆λ1 × · · · × a ∆ l(λ) λl(λ) It has Ehrhart polynomial i(F(lλim,a), t) =.
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