617 research outputs found
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
Distribution of crossings, nestings and alignments of two edges in matchings and partitions
We construct an involution on set partitions which keeps track of the numbers
of crossings, nestings and alignments of two edges.
We derive then the symmetric distribution of the numbers of crossings and
nestings in partitions, which generalizes Klazar's recent result in perfect
matchings. By factorizing our involution through bijections between set
partitions and some path diagrams we obtain the continued fraction expansions
of the corresponding ordinary generating functions.Comment: 12 page
Enumeration of totally positive Grassmann cells
Alex Postnikov has given a combinatorially explicit cell decomposition of the
totally nonnegative part of a Grassmannian, denoted Gr_{kn}+, and showed that
this set of cells is isomorphic as a graded poset to many other interesting
graded posets. The main result of this paper is an explicit generating function
which enumerates the cells in Gr_{kn}+ according to their dimension. As a
corollary, we give a new proof that the Euler characteristic of Gr_{kn}+ is 1.
Additionally, we use our result to produce a new q-analog of the Eulerian
numbers, which interpolates between the Eulerian numbers, the Narayana numbers,
and the binomial coefficients.Comment: 21 pages, 10 figures. Final version, with references added and minor
corrections made, to appear in Advances in Mathematic
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
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