8,710 research outputs found
On k-crossings and k-nestings of permutations
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
International audienceAlignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type , and the cycles of signed permutations are understood in the corresponding bare tableaux of type . 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 in terms of permutation tableaux of type .De nombreuses statistiques importantes des permutations signées sont réalisées dans les tableaux de permutations ou ”bare” tableaux de type correspondants : les alignements, croisements et inversions des permutations signées sont réalisés dans les tableaux de permutations de type correspondants, et les cycles des permutations signées sont comprises dans les ”bare” tableaux de type 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 en termes de tableaux de permutations de type
Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux
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
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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