Abstract: Publication date: Available online 10 January 2020Source: Advances in Applied MathematicsAuthor(s): Dillon Mayhew, Mike Newman, Geoff WhittleAbstractA 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.

Abstract: Publication date: April 2020Source: Advances in Applied Mathematics, Volume 115Author(s): Yuma MizunoAbstractFor 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.

Abstract: Publication date: April 2020Source: Advances in Applied Mathematics, Volume 115Author(s): Lisa Hui SunAbstractIn 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.

Abstract: Publication date: March 2020Source: Advances in Applied Mathematics, Volume 114Author(s): Rahul KumarAbstractRecently Dixit, Kesarwani, and Moll introduced a generalization Kz,w(x) of the modified Bessel function Kz(x) and showed that it satisfies an elegant theory similar to that of Kz(x). In this paper, we show that while K12(x) is an elementary function, K12,w(x) can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function η(z). We also establish a connection between K12,w(x) and the cumulative distribution function corresponding to the Voigt line profile.

Abstract: Publication date: March 2020Source: Advances in Applied Mathematics, Volume 114Author(s): Alexander Ruys de Perez, Laura Felicia Matusevich, Anne ShiuAbstractWe introduce the factor complex of a neural code, and show how intervals and maximal codewords are captured by the combinatorics of factor complexes. We use these results to obtain algebraic and combinatorial characterizations of max-intersection-complete codes, as well as a new combinatorial characterization of intersection-complete codes.

Abstract: Publication date: March 2020Source: Advances in Applied Mathematics, Volume 114Author(s): Benjamin Braun, Marie MeyerAbstractThis paper initiates the study of the Laplacian simplex TG obtained from a finite graph G by taking the convex hull of the columns of the Laplacian matrix for G. Basic properties of these simplices are established, and then a systematic investigation of TG for trees, cycles, and complete graphs is provided. Motivated by a conjecture of Hibi and Ohsugi, our investigation focuses on reflexivity, the integer decomposition property, and unimodality of Ehrhart h⁎-vectors. We prove that if G is a tree, odd cycle, complete graph, or a whiskering of an even cycle, then TG is reflexive. We show that while TKn has the integer decomposition property, TCn for odd cycles does not. The Ehrhart h⁎-vectors of TG for trees, odd cycles, and complete graphs are shown to be unimodal. As a special case it is shown that when n is an odd prime, the Ehrhart h⁎-vector of TCn is given by (h0⁎,…,hn−1⁎)=(1,…,1,n2−n+1,1,…,1). We also provide a combinatorial interpretation of the Ehrhart h⁎-vector for TKn.

Abstract: Publication date: March 2020Source: Advances in Applied Mathematics, Volume 114Author(s): Shi-Mei Ma, Jun Ma, Yeong-Nan YehAbstractIn this paper, we first consider an alternate formulation of the David-Barton identity which relates the alternating run polynomials to Eulerian polynomials. By using this alternate formulation, we see that for any γ-positive polynomial, there exists a David-Barton type identity. We then consider the joint distribution of cycle runs and cycles over the set of permutations. Furthermore, we introduce the definition of semi-γ-positive polynomial. The γ-positivity of a polynomial f(x) is a sufficient (not necessary) condition for the semi-γ-positivity of f(x). We show that the alternating run polynomial of dual Stirling permutations is semi-γ-positive but not γ-positive.

Abstract: Publication date: Available online 6 November 2019Source: Advances in Applied MathematicsAuthor(s): Fan ChungAbstractFor 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.

Abstract: Publication date: Available online 30 September 2019Source: Advances in Applied MathematicsAuthor(s): J.-P. AlloucheAbstractWhat 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.

Abstract: Publication date: Available online 11 July 2019Source: Advances in Applied MathematicsAuthor(s): Irene Pivotto, Gordon RoyleAbstractIn 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.

Abstract: Publication date: Available online 29 May 2019Source: Advances in Applied MathematicsAuthor(s): Fan Chung, Persi Diaconis, Ron GrahamAbstractWe 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.

Abstract: Publication date: Available online 17 May 2019Source: Advances in Applied MathematicsAuthor(s): William Y.C. Chen, Harold R.L. YangAbstractThe 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.

Abstract: Publication date: Available online 2 May 2019Source: Advances in Applied MathematicsAuthor(s): Joseph E. Bonin, Carolyn Chun, Steven D. NobleAbstractVf-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.

Abstract: Publication date: Available online 30 April 2019Source: Advances in Applied MathematicsAuthor(s): Joseph E. Bonin, Carolyn Chun, Steven D. NobleAbstractIn [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.