3,412 research outputs found

    Partial words with a unique position starting a square

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    We consider partial words with a unique position starting a power. We show that over a kk letter alphabet, a partial word with a unique position starting a square can contain at most kk squares. This is in contrast to full words which can contain at most one power if a unique position starts a power. For certain higher powers we exhibit binary partial words containing three powers all of which start at the same position

    Pattern occurrence statistics and applications to the Ramsey theory of unavoidable patterns

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    As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic pattern occurrence statistics and use them in conjunction with the probabilistic method to establish new results about the Ramsey theory of unavoidable patterns in the full word case (both nonabelian sense and abelian sense) and in the partial word case. Keywords: Combinatorics on words; Partial words; Unavoidable patterns; Abelian patterns; Probabilistic method; Analytic combinatorics; Ramsey theory.Comment: 22 pages, 5 figures, presented at SIAM DM 201

    Constructing Words with High Distinct Square Densities

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    Fraenkel and Simpson showed that the number of distinct squares in a word of length n is bounded from above by 2n, since at most two distinct squares have their rightmost, or last, occurrence begin at each position. Improvements by Ilie to 2nΘ(logn)2n-\Theta(\log n) and by Deza et al. to 11n/6 rely on the study of combinatorics of FS-double-squares, when the maximum number of two last occurrences of squares begin. In this paper, we first study how to maximize runs of FS-double-squares in the prefix of a word. We show that for a given positive integer m, the minimum length of a word beginning with m FS-double-squares, whose lengths are equal, is 7m+3. We construct such a word and analyze its distinct-square-sequence as well as its distinct-square-density. We then generalize our construction. We also construct words with high distinct-square-densities that approach 5/6.Comment: In Proceedings AFL 2017, arXiv:1708.0622

    An algorithm to generate exactly once every tiling with lozenges of a domain

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    We first show that the tilings of a general domain form a lattice which we then undertake to decompose and generate without any redundance. To this end, we study extensively the relatively simple case of hexagons and their deformations. We show that general domains can be broken up into hexagon-like parts. Finally we give an algorithm to generate exactly once every element in the lattice of the tilings of a general domain.Comment: 36 pages, 61 figures. To be published in Theoretical Computer Science (special issue on tilings.

    Visibility graphs and deformations of associahedra

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    The associahedron is a convex polytope whose face poset is based on nonintersecting diagonals of a convex polygon. In this paper, given an arbitrary simple polygon P, we construct a polytopal complex analogous to the associahedron based on convex diagonalizations of P. We describe topological properties of this complex and provide realizations based on secondary polytopes. Moreover, using the visibility graph of P, a deformation space of polygons is created which encapsulates substructures of the associahedron.Comment: 18 pages, 16 figure

    Fourier transform on high-dimensional unitary groups with applications to random tilings

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    A combination of direct and inverse Fourier transforms on the unitary group U(N)U(N) identifies normalized characters with probability measures on NN-tuples of integers. We develop the NN\to\infty version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with Law of Large Numbers and Central Limit Theorem for global behavior of corresponding random NN-tuples. As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons. Another application is a central limit theorem for the U(N)U(N) quantum random walk with random initial data.Comment: 66 pages, 10 figure

    Interpolation by periods in planar domain

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    Let ΩR2\Omega\subset\mathbb R^2 be a countably connected domain. To any closed differential form of degree 11 in Ω\Omega with components in L2(Ω)L^2(\Omega) one associates the sequence of its periods around holes in Ω\Omega, that is around bounded connected components of R2Ω\mathbb R^2\setminus \Omega. For which Ω\Omega the collection of such period sequences coincides with 2\ell^2? We give the answer in terms of metric properties of holes in Ω\Omega.Comment: 87 pages, 13 figure

    Singular polynomials of generalized Kasteleyn matrices

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    Kasteleyn counted the number of domino tilings of a rectangle by considering a mutation of the adjacency matrix: a Kasteleyn matrix K. In this paper we present a generalization of Kasteleyn matrices and a combinatorial interpretation for the coefficients of the characteristic polynomial of KK^\ast (which we call the singular polynomial), where K is a generalized Kasteleyn matrix for a planar bipartite graph. We also present a q-version of these ideas and a few results concerning tilings of special regions such as rectangles.Comment: 15 pages, 4 figure

    Tiling with arbitrary tiles

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    Let TT be a tile in Zn\mathbb{Z}^n, meaning a finite subset of Zn\mathbb{Z}^n. It may or may not tile Zn\mathbb{Z}^n, in the sense of Zn\mathbb{Z}^n having a partition into copies of TT. However, we prove that TT does tile Zd\mathbb{Z}^d for some dd. This resolves a conjecture of Chalcraft.Comment: 23 pages, 19 figures; slightly update

    On the Steinhaus tiling problem in three dimensions

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    H. Steinhaus asked in the 1950's whether there exists a set in the plane R^2 meeting every isometric copy of Z^2 in precisely one point. Such a "Steinhaus set" was constructed by Jackson and Mauldin. What about three-space R^3? Is there a subset of R^3 meeting every isometric copy of Z^3 in exactly one point? We offer heuristic evidence that the answer is "no"
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