315 research outputs found

    A closed formula for the number of convex permutominoes

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    In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes

    Enumeration of generalized polyominoes

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    As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular kk-gons. For n≤4n\le 4 we determine formulas for the number ak(n)a_k(n) of generalized polyominoes consisting of nn regular kk-gons. Additionally we give a table of the numbers ak(n)a_k(n) for small kk and nn obtained by computer enumeration. We finish with some open problems for kk-polyominoes.Comment: 10 pages, 6 figures, 3 table

    Evolutionary Dynamics in a Simple Model of Self-Assembly

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    We investigate the evolutionary dynamics of an idealised model for the robust self-assembly of two-dimensional structures called polyominoes. The model includes rules that encode interactions between sets of square tiles that drive the self-assembly process. The relationship between the model's rule set and its resulting self-assembled structure can be viewed as a genotype-phenotype map and incorporated into a genetic algorithm. The rule sets evolve under selection for specified target structures. The corresponding, complex fitness landscape generates rich evolutionary dynamics as a function of parameters such as the population size, search space size, mutation rate, and method of recombination. Furthermore, these systems are simple enough that in some cases the associated model genome space can be completely characterised, shedding light on how the evolutionary dynamics depends on the detailed structure of the fitness landscape. Finally, we apply the model to study the emergence of the preference for dihedral over cyclic symmetry observed for homomeric protein tetramers

    Enumeration of symmetry classes of convex polyominoes on the honeycomb lattice

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    Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the generating series of free (i.e. up to reflection and rotation) and of asymmetric convex hexagonal polyominoes, according to area and half-perimeter. We give explicit formulas or implicit functional equations for the generating series, which are convenient for computer algebra.Comment: 21 pages, 16 figures, 2 tables. This is the full version of a paper presented at the FPSAC Conference in Vancouver, Canada, June 28 -- July 2, 200

    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
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