8,710 research outputs found

    On k-crossings and k-nestings of permutations

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    We introduce k-crossings and k-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of k-noncrossing permutations is equal to the number of k-nonnesting permutations. We also provide some enumerative results for k-noncrossing permutations for some values of k

    Alignments, crossings, cycles, inversions, and weak Bruhat order in permutation tableaux of type BB

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    International audienceAlignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type BB, and the cycles of signed permutations are understood in the corresponding bare tableaux of type BB. We find the relation between the number of alignments, crossings and other statistics of signed permutations, and also characterize the covering relation in weak Bruhat order on Coxeter system of type BB in terms of permutation tableaux of type BB.De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type BB correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type BB correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type BB correspondants. Cela nous mène à relier le nombre d’alignements et de croisements avec d’autres statistiques des permutations signées, et aussi de caractériser la relation de couverture dans l’ordre de Bruhat faible sur des systèmes de Coxeter de type BB en termes de tableaux de permutations de type BB

    Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux

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    We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according to crossings; the polynomial enumerating permutations with a fixed set of descent bottoms according to occurrences of the generalized pattern 2-31.Comment: 37 pages, 4 figures, new references adde

    Combinatorial of restricted permutations according to the number of crossings

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    In this thesis, we introduced and carried out a combinatorial study of permutations that avoid one or two patterns of length 3 according to the statistic number of crossings. For this purpose, we manipulated a bijection of Elizalde and Pak and constructed other bijections that preserve the number of crossings. As results, we found, throughout these bijections, various relationships on the distributions of the number of crossings on restricted permutations as well as combinatorial interpretations in terms of the number of crossings on permutations with forbidden patterns of some well known triangles in the literature.Comment: 61 pages, in French languag
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