arXiv subjects

Gábor Hetyei

Publications and source records attributed to Gábor Hetyei.

At least 19 recordsLinked to original sources

arXiv2026

Generalized formulas for the Jones polynomial of a rational link

We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and Abu-Qamar for the Tutte polynomial of the Tait graph of an alternating diagram of a rational link, as well as the matrix formulas of Lawrence and Rosenstein for the Jones polynomial of a rational link. Our approach uses the colored version of Brylawski's tensor product formula for Tutte polynomials of colored graphs, due to Diao, Hetyei, and Hinson. Furthermore, generalizing the formulas of Qazaqzeh, Yasein, and Abu-Qamar, we present a finite automaton that computes the crossing signs, thereby enabling the calculation of the writhe of a standard diagram of a rational link.

math.GT↗

arXiv2026

On the structure of the d-indivisible noncrossing partition posets

We study the poset of d-indivisible noncrossing partitions introduced by Mühle, Nadeau and Williams. These are noncrossing partitions such that each block and each dual block has cardinality~$1$ modulo $d$. Generalizing the work of Speicher, we introduce a generating function approach to reach new enumerative results and recover some known formulas on the cardinality, the Möbius function and the rank numbers. We compute the antipode of the Hopf algebra of d-indivisible noncrossing partition posets. Generalizing work of Stanley, we show that the bijection between the maximal chains and d-parking functions introduced by Mühle, Nadeau and Williams gives rise to an EL-labeling. Generalizing a construction of Yan, we also introduce d-parking trees which are in bijective correspondence with the maximal chains.

math.CO↗

arXiv2026

The toric g-vector of nestohedra

We show the entries of the toric g-vector of any dual simplicial polytope is a nonnegative integer linear combination of its gamma-vector entries. We give combinatorial interpretations of the toric g-vector of three classical polytopes: the associahedron, the cyclohedron and the permutahedron. Each is expressed in terms of 123-avoidance on a set of structures, namely, parking functions, functions and parking trees. Using our notion of parking trees, we extend our combinatorial model to all chordal nestohedra. As a corollary we obtain combinatorial proofs of nonnegativity of the toric g-vector for all of these polytopes.

math.CO↗

arXiv2026

Hypermaps with hyperedges of length at most $3$

We study the computation of our recently introduced Whitney polynomial and the enumeration of the spanning hypertrees for hypermaps whose hyperedges have length at most $3$. This is a class of hypermaps where the computation of the above invariants depends only on the underlying (multi)hypergraph structure. We develop deletion-contraction formulas involving six types of generalized loops and bridges, and we prove results on special substitutions into our Whitney polynomial. We generalize the reliability polynomial and the random cluster model to hypermaps in general in such a way that they can be computed using our Whitney polynomial. Finally we explicitly count the spanning hypertrees in reciprocals of plane graphs in which every vertex has degree at most $3$.

math.CO↗

arXiv2026

Labeling regions in deformations of graphical arrangements

Combining Carver's variant of the Farkas' lemma with the Flow Decomposition Theorem we show that the regions of any deformation of a graphical arrangement may be bijectively labeled with a set of weighted digraphs containing directed cycles of negative weight only. Bounded regions correspond to strongly connected digraphs. The study of the resulting labelings allows us to add the omitted details in Stanley's proof on the injectivity of the Pak-Stanley labeling of the regions of the extended Shi arrangement, to generalize the ceiling diagrams in the deleted Shi and Ish arrangements studied by Armstrong and Rhoades and to introduce a new labeling of the regions in the Fuss-Catalan arrangement. We also point out that Athanasiadis-Linusson labelings may be used to directly count regions in a class of arrangements properly containing the extended Shi arrangement and the Fuss-Catalan arrangement.

math.CO↗

arXiv2024

A Whitney polynomial for hypermaps

We introduce a Whitney polynomial for hypermaps and use it to generalize the results connecting the circuit partition polynomial to the Martin polynomial and the results on several graph invariants.

math.CO↗

arXiv2024

On reduced unicellular hypermonopoles

The problem of counting unicellular hypermonopoles by the number of their hyperedges is equivalent to describing the cycle length distribution of a product of two circular permutations, first solved by Zagier. The solution of this problem has also been used in the study of the cycle graph model of Bafna and Pevzner and of related models in mathematical biology. In this paper we develop a method to compute the finite number of reduced unicellular hypermonopoles of a given genus. The problem of representing any hypermap as a drawing is known to be simplifiable to solving the same problem for reduced unicellular hypermonopoles. We also outline a correspondence between our hypermap model, the cycle graph model of Bafna and Pevzner, and the polygon gluing model of Alexeev and Zograf. Reduced unicellular hypermonopoles correspond to reduced objects in the other models as well, and the notion of genus is the same.

math.CO↗

arXiv2023

Catalan-Spitzer permutations

We study two classes of permutations intimately related to the visual proof of Spitzer's lemma and Huq's generalization of the Chung-Feller theorem. Both classes of permutations are counted by the Fuss-Catalan numbers. The study of one class leads to a generalization of results of Flajolet from continued fractions to continuants. The study of the other class leads to the discovery of a restricted variant of the Foata--Strehl group action.

math.CO↗

arXiv2022

Spanning hypertrees, vertex tours and meanders

This paper revisits the notion of a spanning hypertree of a hypermap introduced by one of its authors and shows that it allows to shed new light on a very diverse set of recent results. The tour of a map along one of its spanning trees used by Bernardi may be generalized to hypermaps and we show that it is equivalent to a dual tour described by Cori and Mach\`ı. We give a bijection between the spanning hypertrees of the reciprocal of the plane graph with $2$ vertices and $n$ parallel edges and the meanders of order $n$ and a bijection of the same kind between semimeanders of order $n$ and spanning hypertrees of the reciprocal of a plane graph with a single vertex and $n/2$ nested edges. We introduce hyperdeletions and hypercontractions in a hypermap which allow to count the spanning hypertrees of a hypermap recursively, and create a link with the computation of the Tutte polynomial of a graph. Having a particular interest in hypermaps which are reciprocals of maps, we generalize the reduction map introduced by Franz and Earnshaw to enumerate meanders to a reduction map that allows the enumeration of the spanning hypertrees of such hypermaps.

math.CO↗

arXiv2020

The type $B$ permutohedron and the poset of intervals as a Tchebyshev transform

We show that the order complex of intervals of a poset, ordered by inclusion, is a Tchebyshev triangulation of the order complex of the original poset. Besides studying the properties of this transformation, we show that the dual of the type $B$ permutohedron is combinatorially equivalent to the suspension of the order complex of the poset of intervals of a Boolean algebra (with the minimum and maximum elements removed).

math.CO↗

arXiv2020

Classification of uniform flag triangulations of the boundary of the full root polytope of type $A$

The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.

math.CO↗

arXiv2019

Alternation acyclic tournaments

We define a tournament to be alternation acyclic if it does not contain a cycle in which descents and ascents alternate. Using a result by Athanasiadis on hyperplane arrangements, we show that these tournaments are counted by the median Genocchi numbers. By establishing a bijection with objects defined by Dumont, we show that alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind. Unexpected consequences of our results include a pair of ordinary generating function formulas for the Genocchi numbers of both kinds and a new very simple model for the normalized median Genocchi numbers.

math.CO↗

arXiv2018

The Braid Index Of Reduced Alternating Links

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid indices equal to the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if $D$ is a reduced alternating link diagram of an alternating link $L$, then $b(L)$, the braid index of $L$, equals the number of Seifert circles in $D$ if and only if $G_S(D)$ contains no edges of weight one. Here $G_S(D)$, called the Seifert graph of $D$, is an edge weighted simple graph obtained from $D$ by identifying each Seifert circle of $D$ as a vertex of $G_S(D)$ such that two vertices in $G_S(D)$ are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in $D$ and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well known MFW inequality, which states that the $a$-span of the HOMFLY polynomial of $L$ is a lower bound of $2b(L)-2$, as well as a result due to Yamada, which states that the minimum number of Seifert circles over all link diagrams of $L$ equals $b(L)$.

math.GT↗

arXiv2017

Counting partitions of a fixed genus

We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus $2$ partitions, after identifying all genus $2$ primitive partitions, using a computer-assisted search.

math.CO↗

arXiv2016

The HOMFLY Polynomial of Links in Closed Braid Form

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an $n$-string closed braid diagram that is also reduced and alternating, is exactly $n$. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

math.GT↗

arXiv2016

Efron's coins and the Linial arrangement

We characterize the tournaments that are dominance graphs of sets of (unfair) coins in which each coin displays its larger side with greater probability. The class of these tournaments coincides with the class of tournaments whose vertices can be numbered in a way that makes them semiacyclic, as defined by Postnikov and Stanley. We provide an example of a tournament on nine vertices that can not be made semiacyclic, yet it may be represented as a dominance graph of coins, if we also allow coins that display their smaller side with greater probability. We conclude with an example of a tournament with $81$ vertices that is not the dominance graph of any system of coins.

math.CO↗

arXiv2015

Generalized Tchebyshev triangulations

After fixing a triangulation $L$ of a $k$-dimensional simplex that has no new vertices on the boundary, we introduce a triangulation operation on all simplicial complexes that replaces every $k$-face with a copy of $L$, via a sequence of induced subdivisions. The operation may be performed in many ways, but we show that the face numbers of the subdivided complex depend only on the face numbers of the original complex, in a linear fashion. We use this linear map to define a sequence of polynomials generalizing the Tchebyshev polynomials of the first kind and show, that in many cases, but not all, the resulting polynomials have only real roots, located in the interval $(-1,1)$. Some analogous results are shown also for generalized Tchebyshev polynomials of the higher kind, defined using summing over links of all original faces of a given dimension in our generalized Tchebyshev triangulations. Generalized Tchebyshev triangulations of the boundary complex of a cross-polytope play a central role in our calculations, and for some of these we verify the validity of a generalized lower bound conjecture by the second author.

math.CO↗

arXiv2013

Counting genus one partitions and permutations

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides a new way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations.

math.CO↗