7,353 research outputs found

    Ten Conferences WORDS: Open Problems and Conjectures

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    In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

    Some families of increasing planar maps

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    Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with 2n2n faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by n1/2n^{1/2}, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by (6/11)logn(6/11)\log n converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations

    Transition Property For Cube-Free Words

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    We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair (u,v)(u,v) of dd-ary cube-free words, if uu can be infinitely extended to the right and vv can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word ww over the same alphabet such that uwvuwv is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.Comment: 14 pages, 5 figure
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