Feasible Bases for a Polytope Related to the Hamilton Cycle Problem

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

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

In this paper, we consider the volume of a special kind of flow polytope. We show that its volume satisfies a certain system of differential equations, and conversely, the solution of the system of differential equations is unique up to a constant multiple. In addition, we give an inductive formula for the volume with respect to the rank of the root system of type $A$.

Volumes of Flow Polytopes Related to Caracol Graphs

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

Flow Polytopes and the Space of Diagonal Harmonics

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 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.

$h$-Polynomials via Reduced Forms

The flow polytope $F_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $F_{\widetilde{G}}$ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial $\prod_{(i,j)\in E(G)}x_{ij}$. We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but also regular flag triangulations of flow polytopes. We prove that reduced forms in the subdivision algebra are generalizations of $h$-polynomials of the triangulations of flow polytopes. We deduce several corollaries of the above results, most notably proving certain cases of a conjecture of Kirillov about the nonnegativity of reduced forms in the noncommutative quasi-classical Yang-Baxter algebra.

The polytope of Tesler matrices

26 pages, 4 figures. v2 has typos fixed, updated references, and a final remarks section including remarks from previous sections International audience We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. We show that $Tes_n(a)$ is a flow polytope and therefore the number of Tesler matrices is counted by the type $A_n$ Kostant partition function evaluated at $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of $Tes_n(a)$ when all $a_i>0$ is given by the Mahonian numbers and calculate the volume of $Tes_n(1,1,...,1)$ to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape. On présente le polytope de Tesler $Tes_n(a)$, dont les points réticuilaires sont les matrices de Tesler de taillen avec des sommes des équerres $a_1,a_2,...,a_n in Z_{\geq 0}$. On montre que $Tes_n(a)$ est un polytope de flux. Donc lenombre de matrices de Tesler est donné par la fonction de Kostant de type An évaluée à ($(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$On décrit les faces de ce polytope en termes de “tableaux de Tesler” et on caractérise quand le polytope est simple.On montre que l’h-vecteur de $Tes_n(a)$ , quand tous les $a_i>0$ , est donnée par le nombre de permutations avec unnombre donné d’inversions et on calcule le volume de T$Tes_n(1,1,...,1)$ comme un produit de nombres de Catalanconsécutives multiplié par le nombre de tableaux standard de Young en forme d’escalier