54,239 research outputs found

    Stacked polytopes and tight triangulations of manifolds

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    Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here we present other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well. One of the conditions is that all vertex links are stacked spheres, which implies that the triangulation is in Walkup's class K(d)\mathcal{K}(d). We show that in any dimension d≥4d\geq 4 \emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz are tight. Furthermore, triangulations with kk-stacked vertex links and the centrally symmetric case are discussed.Comment: 28 pages, 2 figure

    Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

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    We deliver here second new H(x)−binomials′\textit{H(x)}-binomials' recurrence formula, were H(x)−binomials′H(x)-binomials' array is appointed by Ward−HoradamWard-Horadam sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of p,q−binomialp,q-binomial coefficients onto q−binomialq-binomial coefficients interpretations thus bringing us back to Gyo¨rgyPoˊlyaGy{\"{o}}rgy P\'olya and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

    On the expected number of perfect matchings in cubic planar graphs

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    A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with nn vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγnc\gamma^n, where c>0c>0 and γ∼1.14196\gamma \sim 1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure

    Crystal constructions in Number Theory

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    Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics
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