本文定义格路拟阵，研究其枚举性质及 Tutte 多项式、特征多项式与 beta 不变量，并证明 Tutte 多项式可多项式时间计算。
摘要：
固定两条从 (0,0) 到 (m,r)、使用东步和北步且 P 始终不越过 Q 上方的格路 P 和 Q。我们证明，从 (0,0) 到 (m,r) 并保持在由 P 与 Q 界定的区域内的格路，可以等同于一类特殊横截拟阵的基，我们将其称为格路拟阵。我们考察这些拟阵的多种枚举方面，并研究三个重要的拟阵不变量，即 Tutte 多项式，以及对于特殊类型格路拟阵的特征多项式和 beta 不变量。特别地，我们证明 Tutte 多项式是两个基本格路统计量的生成函数，并证明某些格路拟阵序列会产生相应的 Tutte 多项式序列，而这些序列具有相对简单的生成函数。我们证明格路拟阵的 Tutte 多项式可在多项式时间内计算。此外，通过分析某些格路拟阵的 beta 不变量，我们还得到一个关于格路的新结果。
英文摘要：
Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the beta invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the beta invariant of certain lattice path matroids.
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 relate matroid connectivity to Tutte-connectivity in an infinite graph. Moreover, we show that the two cycle matroids, the finite-cycle matroid and the cycle matroid, in which also infinite cycles are taken into account, have the same connectivity function. As an application we re-prove that, also for infinite graphs, Tutte-connectivity is invariant under taking dual graphs.
The inner Tutte group of a matroid is a finitely generated abelian group introduced as an algebraic counterpart of Tutte's homotopy theory of matroids. The aim of this work is to provide a new presentation for this group with a set of generators that is smaller than those previously known.
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.
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 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.
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 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.
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…
The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engineering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some …
In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants.
We define and study the Tutte polynomial of a hyperplane arrangement. We introduce a method for computing it by solving an enumerative problem in a finite field. For specific arrangements, the computation of Tutte polynomials is then reduced to certain related enumerative questions. As a consequence, we obtain new formulas for the generating functions enumerating alternating trees, labelled trees, semiorders and Dyck paths.
Specializing the $γ$-basis for the vector space $\mathcal{G}(n,r)$ spanned by the set of symbols on bit sequences with $r$ $1$'s and $n-r$ $0$'s, we obtain a frame or spanning set for the vector space $\mathcal{T}(n,r)$ spanned by Tutte polynomials of matroids having rank $r$ and size $n$. Every Tutte polynomial can be expanded as a linear combination with non-negative integer coefficients of elements in this frame. We give explicit formulas for the elements in this frame. These formulas combine to give an expansion of the Tutte polynomial with coefficients obtained by summing numerical invari…
We consider a specialization $Y_M(q,t)$ of the Tutte polynomial of a matroid $M$ which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of $M$. We show that the coefficients of $Y_M(1-p,t)$ are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of $M$. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid.
Lopéz de Medrano-Rinćon-Shaw defined Chern-Schwartz-MacPherson cycles for an arbitrary matroid $M$ and proved by an inductive geometric argument that the unsigned degrees of these cycles agree with the coefficients of $T(M;x,0)$, where $T(M;x,y)$ is the Tutte polynomial associated to $M$. Ardila-Denham-Huh recently utilized this interpretation of these coefficients in order to demonstrate their log-concavity. In this note we provide a direct calculation of the degree of a matroid Chern-Schwartz-MacPherson cycle by taking its stable intersection with a generic tropical linear space of the appro…
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 …
The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohe…
The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a co…
Vertigan has shown that if $M$ is a binary matroid, then $|T_M(-ι,ι)|$, the modulus of the Tutte polynomial of $M$ as evaluated in $(-ι, ι)$, can be expressed in terms of the bicycle dimension of $M$. In this paper, we describe how the argument of the complex number $T_M(-ι,ι)$ depends on a certain $\zfour$-valued quadratic form that is canonically associated with $M$. We show how to evaluate $T_M(-ι,ι)$ in polynomial time, as well as the canonical tripartition of $M$ and further related invariants.
A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHP…
The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is related to previous works of Cameron and Fink and of Kálmán and Postnikov. We then define the universal Tutte polynomial $\T_n$, which is a polynomial of degree $n$ in $2+(2^n-1)$ variables that specializes to the Tutte polynomials of all polymatroids (hence all matroids) on a ground set with $n$ elements. The universal polynomial $\T_n$ admits three kinds of sy…
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.
We introduce the ``trivariate Tutte polynomial" of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the …
In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene's theorem, thereby expressing harmonic weight enumerators of codes as evaluations of harmonic Tutte polynomials.
Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a coboundary condition (that $f-g$ is a flow over $A$ relative to $G$) and a congruence condition (that the size of the supports of $f$ and $g$ satisfy some congruence condition modulo an integer). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals an evaluation of the Tutte polynomial of $M(G…
We introduce and study filtrations of a matroid on a linearly ordered ground set, which are particular sequences of nested sets. A given basis can be decomposed into a uniquely defined sequence of bases of minors, such that these bases have an internal/external activity equal to 1/0 or 0/1 (in the sense of Tutte polynomial activities). This decomposition, which we call the active filtration/partition of the basis, refines the known partition of the ground set into internal and external elements with respect to a given basis. It can be built by a certain closure operator, which we call the acti…
Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families of new identities for antimatroids, including trees, posets, chordal graphs and finite point sets in $\mathbb{R}^n$. It also gives a "new" linear relation for matroids that is implied by Brylawski's identities.
We transform Tutte-Grothedieck invariants thus also Tutte polynomials on matroids so that the contraction-deletion rule for loops (isthmuses) coincides with the general case.
Using a quantum field theory renormalization group-like differential equation, we give a new proof of the recipe theorem for the Tutte polynomial for matroids. The solution of such an equation is in fact given by some appropriate characters of the Hopf algebra of isomorphic classes of matroids, characters which are then related to the Tutte polynomial for matroids. This Hopf algebraic approach also allows to prove, in a new way, a matroid Tutte polynomial convolution formula appearing in W. Kook {\it et. al., J. Comb. Series} {\bf B 76} (1999).
The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite famili…
We generalize the Tutte polynomial of a matroid to a morphism of matroids via the K-theory of flag varieties. We introduce two different generalizations, and demonstrate that each has its own merits, where the trade-off is between the ease of combinatorics and geometry. One generalization recovers the Las Vergnas Tutte polynomial of a morphism of matroids, which admits a corank-nullity formula and a deletion-contraction recursion. The other generalization does not, but better reflects the geometry of flag varieties.
The Tutte polynomial of a connected graph was originally defined by Tutte as a sum over all spanning trees of monomials depending on a fixed linear order on the set of edges. Tuttle proved that while these monomials do depend on the linear order, the sum does not. The present paper is a result of a reflection upon this classical theorem of Tutte. It is devoted to an elementary and detailed proof of this theorem in its natural generality, i.e. not for graphs, but for matroids. In contrast with usual methods, the emphasis is on the properties of maps (as opposed to elements) naturally associated…
Orlik-Solomon algebras determine the Tutte polynomials of matroids of rank at most three
Xiamen University（厦门大学）；Qinghai Minzu University（青海民族大学）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）；Guangdong Polytechnic Normal University（广东技术师范大学）
In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials encompasses invariants defined for CW complexes by Beck-Breuer-Godkin-Martin and Duval-Klivans-Martin. Furthermore, we answer a question by Bajo-Burdick-Chmutov, concerning the modified Tutte-Krushkal-Renhardy polynomials defined by these authors: to this end, we prove that the product of two arithmetic multiplicity functions on a matroid is again an arithmetic multiplicity function.
The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial $\mathcal{J}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first prove that $\mathcal{J}_{P}(x,t)$ and $\mathcal{J}_{P}(t,y)$ are interpolating for any fixed real number $t\geq 1$, and then we study the coefficients of high-order terms in $\mathcal{J}_{P}(x,1)$ and $\mathcal{J}_{P}(1,y)$. These results generalize results on interior and exterior polynomials of hypergraphs.
We give a quantum-inspired $O(n^4)$ algorithm computing the Tutte polynomial of a lattice path matroid, where $n$ is the size of the ground set of the matroid. Furthermore, this can be improved to $O(n^2)$ arithmetic operations if we evaluate the Tutte polynomial on a given input, fixing the values of the variables. The best existing algorithm, found in 2004, was $O(n^5)$, and the problem has only been known to be polynomial time since 2003. Conceptually, our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems s…
The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this compu…
Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of M(IA(G)) yield the interlace polynomials of G, introduced by Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584]. Interlace polynomials subsequently introduced by other authors may be obtained from parametrized Tutte polynomials of the binary matroid represented by (I | A(G) | I+A(G)).
Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to ind…
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $τ(x,y)$ of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution $x \to (x-1),\; y \to (y-1)$ in their rank generating functions $ρ(x,y)$. The $q$-analogue of the passage from a Tutte polynomial to its corresponding RGF is straight-forward, but the analogue of the reverse process $x \to (x-1),\; y \to (y-1)$ is more de…
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobas and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial. We show that our Tutte polynomials of…
Using Tutte's combinatorial definition of a map we define a $Δ$-matroid purely combinatorially and show that it is identical to Bouchet's topological definition.
The Tutte equations are ported (or set-pointed) when the equations F(N) = g_e F(N/e) + r_e F(N\e) are omitted for elements e in a distinguished set called ports. Solutions F can distinguish different orientations of the same matroid. A ported extensor with ground set is a decomposible element in the exterior algebra over a vector space with a given basis, called the ground set, containing a distinguished subset called ports. These can represent ported matroids and have analogous dualization, deletion and contraction operations. A ported extensor function is defined using dualization, port elem…
In this paper we give a new proof of the universality of the Tutte polynomial for matroids. This proof uses appropriate characters of Hopf algebra of matroids, algebra introduced by Schmitt (1994). We show that these Hopf algebra characters are solutions of some differential equations which are of the same type as the differential equations used to describe the renormalization group flow in quantum field theory. This approach allows us to also prove, in a different way, a matroid Tutte polynomial convolution formula published by Kook, Reiner and Stanton (1999). This FPSAC contribution is an ex…
