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  • Asymptotic enumeration of linear hypergraphs with given number of vertices and edges
    Adv. Appl. Math. (IF 1.008) Pub Date : 2020-01-16
    Brendan D. McKay; Fang Tian

    For n≥3, let r=r(n)≥3 be an integer. A hypergraph is r-uniform if each edge is a set of r vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear r-uniform hypergraphs on n→∞ vertices is determined asymptotically when the number of edges is m(n)=o(r−3n3/2). As one application, we find the probability of linearity for the independent-edge model of random r-uniform hypergraph when the expected number of edges is o(r−3n3/2). We also find the probability that a random r-uniform linear hypergraph with a given number of edges contains a given subhypergraph.

    更新日期:2020-01-17
  • On the joint distribution of cyclic valleys and excedances over conjugacy classes of Sn
    Adv. Appl. Math. (IF 1.008) Pub Date : 2020-01-15
    M. Crossan Cooper; William S. Jones; Yan Zhuang

    We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the γ-positivity of the excedance number distribution over any fixed conjugacy class along with a combinatorial interpretation of the γ-coefficients.

    更新日期:2020-01-15
  • Fractal classes of matroids
    Adv. Appl. Math. (IF 1.008) Pub Date : 2020-01-10
    Dillon Mayhew; Mike Newman; Geoff Whittle

    A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k≥5) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

    更新日期:2020-01-11
  • An extension of the Andrews–Warnaar partial theta function identity
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-12-19
    Lisa Hui Sun

    In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews–Warnaar partial theta function identity, and also unifies several results on partial theta functions due to Ramanujan, Kim and Lovejoy. We also establish a two-term version of the extension, which can be used to derive identities for partial and false theta functions. Finally, we present a relation between the big q-Jacobi polynomials and the Andrews–Warnaar partial theta function identity.

    更新日期:2020-01-04
  • Jacobian matrices of Y-seed mutations
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-12-20
    Yuma Mizuno

    For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the C-matrix of the cluster transformation in a certain limit of cluster variables. As an application, we prove that quivers associated with once-punctured surfaces do not have maximal green or reddening sequences.

    更新日期:2020-01-04
  • Regularity lemmas for clustering graphs
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-11-06
    Fan Chung

    For a graph G with a positive clustering coefficient C, it is proved that for any positive constant ϵ, the vertex set of G can be partitioned into finitely many parts, say S1,S2,…,Sm, such that all but an ϵ fraction of the triangles in G are contained in the projections of tripartite subgraphs induced by (Si,Sj,Sk) which are ϵ-Δ-regular, where the size m of the partition depends only on ϵ and C. The notion of ϵ-Δ-regular, which is a variation of ϵ-regular for the original regularity lemma, concerns triangle density instead of edge density. Several generalizations and variations of the regularity lemma for clustering graphs are derived.

    更新日期:2020-01-04
  • The zeta-regularized product of odious numbers
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-09-30
    J.-P. Allouche

    What is the product of all odious integers, i.e., of all integers whose binary expansion contains an odd number of 1's? Or more precisely, how to define a product of these integers which is not infinite, but still has a “reasonable” definition? We will answer this question by proving that this product is equal to π1/42φe−γ, where γ and φ are respectively the Euler-Mascheroni and the Flajolet-Martin constants.

    更新日期:2020-01-04
  • The Möbius matroids
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-07-11
    Irene Pivotto; Gordon Royle

    In three influential papers in the 1980s and early 1990s, Joe Kung laid the foundations for extremal matroid theory which he envisaged as finding the growth rate of certain classes of matroids along with a characterisation of the extremal matroids in each such class. At the time, he was particularly interested in the minor-closed classes of binary matroids obtained by excluding the cycle matroids of the Kuratowski graphs K3,3 and/or K5. While he obtained strong bounds on the growth rate of these classes, it seems difficult to give the exact growth rate without a complete characterisation of the matroids in each class, which at the time seemed hopelessly complicated. Many years later, Mayhew, Royle and Whittle gave a characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor, from which the answers to Kung's questions follow immediately. In this characterisation, two thin families of binary matroids play an unexpectedly important role as the only non-cographic infinite families of internally 4-connected binary matroids with no M(K3,3)-minor. As the matroids are closely related to the cubic and quartic Möbius ladders, they were called the triangular Möbius matroids and the triadic Möbius matroids. Preliminary investigations of the class of binary matroids with no M(K5)-minor suggest that, once again, the triangular Möbius matroids will be the extremal internally 4-connected matroids in this class. Here we undertake a systematic study of these two families of binary matroids collecting in one place fundamental information about them, including their representations, connectivity properties, minor structure, automorphism groups and their chromatic polynomials. Along the way, we highlight the different ways in which these matroids have arisen naturally in a number of results and problems (both open and settled) in structural and extremal matroid theory.

    更新日期:2020-01-04
  • Permanental generating functions and sequential importance sampling
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-05-29
    Fan Chung; Persi Diaconis; Ron Graham

    We introduce techniques for deriving closed form generating functions for enumerating permutations with restricted positions keeping track of various statistics. The method involves evaluating permanents with variables as entries. These are applied to determine the sample size required for a novel sequential importance sampling algorithm for generating random perfect matchings in classes of bipartite graphs.

    更新日期:2020-01-04
  • A context-free grammar for the Ramanujan-Shor polynomials
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-05-17
    William Y.C. Chen; Harold R.L. Yang

    The polynomials ψk(r,x) were introduced by Ramanujan. Berndt, Evans and Wilson obtained a recurrence relation for ψk(r,x). Shor introduced polynomials related to improper edges of a rooted tree, leading to a refinement of Cayley's formula. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation of Berndt, Evans and Wilson. These polynomials also arise in the work of Wang and Zhou on the orbifold Euler characteristics of the moduli spaces of stable curves. Dumont and Ramamonjisoa found a context-free grammar G to generate the number of rooted trees on n vertices with k improper edges. Based on the grammar G, we find a grammar H for the Ramanujan-Shor polynomials. This leads to a formal calculus for these polynomials. In particular, we obtain a grammatical derivation of the Berndt-Evans-Wilson-Shor recursion. We also provide a grammatical approach to the Abel identities and a grammatical explanation of the Lacasse identity.

    更新日期:2020-01-04
  • The excluded 3-minors for vf-safe delta-matroids
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-05-02
    Joseph E. Bonin; Carolyn Chun; Steven D. Noble

    Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids and ribbon-graphic delta-matroids by finding the minimal obstructions, called excluded 3-minors, to membership in the class. We find the unique (up to twisted duality) excluded 3-minor within the class of set systems for the class of vf-safe delta-matroids. In the literature, binary delta-matroids appear in many different guises, with appropriate notions of minor operations equivalent to that of 3-minors, perhaps most notably as graphs with vertex minors. We give a direct explanation of this equivalence and show that some well-known results may be expressed in terms of 3-minors.

    更新日期:2020-01-04
  • Delta-matroids as subsystems of sequences of Higgs lifts
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-04-30
    Joseph E. Bonin; Carolyn Chun; Steven D. Noble

    In [30], Tardos studied special delta-matroids obtained from sequences of Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which all of our results revolve. We give an excluded-minor characterization of the class of full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar characterizations of two other minor-closed classes of delta-matroids that we define using Higgs lifts. We introduce a minor-closed, dual-closed class of Higgs lift delta-matroids that arise from lattice paths. It follows from results of Bouchet that all delta-matroids can be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address which feasible sets can be removed, we give an excluded-minor characterization of delta-matroids within the more general structure of set systems. Many of these excluded minors occur again when we characterize the delta-matroids in which the collection of feasible sets is the union of the collections of bases of matroids of different ranks, and yet again when we require those matroids to have special properties, such as being paving.

    更新日期:2020-01-04
  • RECOVERING A TREE FROM THE LENGTHS OF SUBTREES SPANNED BY A RANDOMLY CHOSEN SEQUENCE OF LEAVES.
    Adv. Appl. Math. (IF 1.008) Pub Date : 2018-09-18
    Steven N Evans,Daniel Lanoue

    Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let Wk , 2 ≤ k ≤ n, be the length of the subtree spanned by the first k leaves. We consider the question, "Can T be identified (up to isomorphism) by the joint probability distribution of the random vector (W2, …, Wn )?" We show that if T is known a priori to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted trees, and certain families with equal weights on all edges such as (k + 1)-valent and rooted k-ary trees for k ≥ 2 and caterpillars.

    更新日期:2019-11-01
  • ENUMERATION OF LONELY PAIRS OF GENE TREES AND SPECIES TREES BY MEANS OF ANTIPODAL CHERRIES.
    Adv. Appl. Math. (IF 1.008) Pub Date : 2019-04-16
    Noah A Rosenberg

    In mathematical phylogenetics, given a rooted binary leaf-labeled gene tree topology G and a rooted binary leaf-labeled species tree topology S with the same leaf labels, a coalescent history represents a possible mapping of the list of gene tree coalescences to the associated branches of the species tree on which those coalescences take place. For certain families of ordered pairs (G, S), the number of coalescent histories increases exponentially or even faster than exponentially with the number of leaves n. Other pairs have only a single coalescent history. We term a pair (G, S) lonely if it has only one coalescent history. Here, we characterize the set of all lonely pairs (G, S). Further, we characterize the set of pairs of rooted binary unlabeled tree shapes at least one of the labelings of which is lonely. We provide formulas for counting lonely pairs and pairs of unlabeled tree shapes with at least one lonely labeling. The lonely pairs provide a set of examples of pairs (G, S) for which the number of compact coalescent histories-which condense coalescent histories into a set of equivalence classes-is equal to the number of coalescent histories. Application of the condition that characterizes lonely pairs can also be used to reduce computation time for the enumeration of coalescent histories.

    更新日期:2019-11-01
  • Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems.
    Adv. Appl. Math. (IF 1.008) Pub Date : 2010-02-18
    Murad Banaji,Gheorghe Craciun

    In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions for injectivity of chemical reaction systems. After developing a translation between the two formalisms, we show that a graph-theoretic test developed earlier in the context of systems with mass action kinetics, can be applied to reaction systems with arbitrary kinetics. The test, which is easy to implement algorithmically, and can often be decided without the need for any computation, rules out the possibility of multiple equilibria for the systems in question.

    更新日期:2019-11-01
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