For a finite multigraph G, the reliability function of G is the probability R_G(q) that if each edge of G is deleted independantly with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if the complex number q is such that R_G(q)=0 then |q|<=1. We verify that this conjectured property of R_G(q) holds if G is a series-parallel network. The proof is by an application of the Hermite-Biehler Theorem and development of a theory of higher-order interlacing for polynomials with only real nonpositive zeros. We conclude by establishing some new inequalities which are satisfied by the f-vector of any matroid without coloops, and by discussing some stronger inequalities which would follow (in the cographic case) from the Brown-Colbourn Conjecture, and are hence true for cographic matroids of series-parallel networks.
The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield, whose properties need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was recently conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, which provide…
In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive formula for the Ehrhart polynomials of $(a,b)$-Catalan matroids. Ferroni showed that uniform and minimal matroids are Ehrhart positive. We show that all sparse paving Schubert matroids are Ehrhart positive and their Ehrhart polynomials are coefficient-wisely bounded by those of minimal and uniform matroids. This confirms a conjecture of Ferroni for the case of s…
In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups, Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig polynomials for matroids. It is known that both the classical Kazhdan-Lusztig polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the inverse of Kazhdan-Lusztig-Stanley functions and define the inverse Kazhdan-Lusztig polynomials for matroids. We also compute these polynomi…
We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.
Following the work of Gao and Xie in [2], we state some properties of the inverse Kazhdan-Lusztig polynomial of a matroid. We also give partial answers to a conjecture that states that regular connected matroids are non-degenerate. We link the degeneracy of a matroid to the inverse Kazhdan-Lusztig polynomial and we show that the Conjecture holds for modular matroids, by proving that degenerate modular matroids are not regular.
We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than graph-theoretical arguments, we find that various known results about these polynomials, including their recursive relations, are both more efficiently and more generally obtained. In addition, we obtain several interrelationships and results for polynomials on multimatroids and delta-matroids that correspond to new interrelationships and results for the corresponding g…
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type invariant of links in the 3-sphere.
The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan-Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule.
Let $ρ$ be a non-negative integer. A $ρ$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $ρ$ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of $ρ$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.
We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^*$-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $h^*$-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality ar…
Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids
School of Mathematical Sciences, Tianjin University of Technology（天津理工大学数学科学学院）；College of Mathematical Sciences & Institute of Mathematics and Interdisciplinary Sciences, Tianjin Normal University（天津师范大学数学科学学院与数学与交叉科学研究院）
We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of a thagomizer matroid.
Richter-Gebert proved that every non-Euclidean uniform oriented matroid admits a biquadratic final polynomial. We extend this result to the non-uniform case.
A catalogue of all non-isomorphic simple connected regular matroids ${\cal M}$ of cardinality $n \leq 15$ is provided on the net. These matroids are given as binary matrix matroids and are sieved from the large pool of all non-isomorphic binary matrix matroids of cardinality $\leq 15$. For each ${\cal M}$ its Tutte polynomial is determined by an algorithm based on internal and external base activity.
Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded minimal free resolutions of the Stanley-Reisner ideals of $M$ and so-called elongations of $M$. Generalizing Greene's theorem from coding theory, we show that the enumerator of a matroid is equivalent to its Tutte polynomial.
Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sence for general delta-matroids. In thi…
We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the Z-polynomial in which the symmetry is a manifestation of Poincare duality.
In contrast to matroids, vf-safe delta-matroids have three kinds of minors and are closed under the operations of twist and loop complementation. We show that the delta-matroids representable over GF(4) with respect to the nontrivial automorphism of GF(4) form a subclass of the vf-safe delta-matroids closed under twist and loop complementation. In particular, quaternary matroids are vf-safe. Using this result, we show that the matroid of a bicycle space of a quaternary matroid M is obtained from M by using loop complementation. As a consequence, the matroid of a bicycle space of a quaternary m…
We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.
We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h^*-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity.
Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan-Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan-Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling …
We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；College of Science, Tianjin University of Technology（天津工业大学理学院）
The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections between geometric duals of plane graphs and duals of matroids. We obtain analogous connections for various types of duality in the literature for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality on ribbon graphs to loop com…
We study the way in which equivariant Kazhdan-Lusztig polynomials, equivariant inverse Kazhdan-Lusztig polynomials, and equivariant Z-polynomials of matroids change under the operation of relaxation of a collection of stressed hyperplanes. This allows us to compute these polynomials for arbitrary paving matroids, which we do in a number of examples, including various matroids associated with Steiner systems that admit actions of Mathieu groups.
In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. As applications we obtain two new infinite families of matroids verifying a conjecture of De Loera et.~al. and present an explicit formula of the Ehrhart polynomial for one of them.
Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding $h^*$-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhar…
We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power $q$ and constant $k$, the characteristic polynomial of any loopless, $GF(q)$-representable matroid with tree-width $k$ has no real zero greater than $q^{k-1}$.
In this note we generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes.
We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activities based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás-Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.
It is known that a lattice path matroid polytope can be associated with two given noncrossing lattice paths on $\mathbb{Z}\times\mathbb{Z}$ with the same end points. In this short note we give explicit formulae for the $f$-vector, toric $f$- and $g$-polynomials of a lattice path matroid polytope when two boundary paths enclose a hook shape.
Inverse Kazhdan--Lusztig polynomials of fan matroids
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；Shenzhen Research Institute of Northwestern Polytechnical University（西北工业大学深圳研究院）
本文研究扇形 matroid 的逆 Kazhdan--Lusztig 多项式和逆 Z 多项式，推导了生成函数并得出显式公式，证明了逆 Kazhdan--Lusztig 多项式系数序列的对数凹性。
We show how the Tutte polynomial of a matroid $M$ can be computed from its condensed configuration, which is a statistic of its lattice of cyclic flats. The results imply that the Tutte polynomial of $M$ is already determined by the abstract lattice of its cyclic flats together with their cardinalities and ranks. They furthermore generalize a similiar statement for perfect matroid designs due to Mphako and help to understand families of matroids with identical Tutte polynomial as constructed by Ken Shoda.
We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the …
We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientat…
In 1977, Yu. V. Matiyasevich proposed a formula expressing the chromatic polynomial of an arbitrary graph as a linear combination of flow polynomials of subgraphs of the original graph. In this paper, we prove that this representation is a particular case of one (easily verifiable) formula, namely, the representation of the characteristic polynomial of an arbitrary matroid as a linear combination of characteristic polynomials of dual matroids. As an application, we represent the flow polynomial of a complete graph with $n$ vertices as the sum of elementary terms with respect to all partitions …
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization of the Stanley's one. Our goal is to extend Stanley's chromatic polynomial to embedded graphs. In contrast to well-known extensions of, say, the Tutte polynomial from abstract to embedded graphs [5], we do not treat an embedded graph as an abstract graph endowed with additional information about the embedding. Instead, we consider the binary delta-matroid a…
We introduce the notion of an arithmetic matroid, whose main example is given by a list of elements of a finitely generated abelian group. In particular we study the representability of its dual, providing an extension of the Gale duality to this setting. Guided by the geometry of generalized toric arrangements, we provide a combinatorial interpretation of the associated arithmetic Tutte polynomial, which can be seen as a generalization of Crapo's formula for the classical Tutte polynomial.
This paper studies systems of polynomial equations that provide information about orientability of matroids. First, we study systems of linear equations over GF(2), originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determine weak orientability for all matroids on at most nine elements and all matroids between ten and twelve elements having rank three. Our experiments indicate that for small rank, about half the time, wh…
