128 research outputs found
Generating trees for permutations avoiding generalized patterns
We construct generating trees with one, two, and three labels for some
classes of permutations avoiding generalized patterns of length 3 and 4. These
trees are built by adding at each level an entry to the right end of the
permutation, which allows us to incorporate the adjacency condition about some
entries in an occurrence of a generalized pattern. We use these trees to find
functional equations for the generating functions enumerating these classes of
permutations with respect to different parameters. In several cases we solve
them using the kernel method and some ideas of Bousquet-M\'elou. We obtain
refinements of known enumerative results and find new ones.Comment: 17 pages, to appear in Ann. Com
The most and the least avoided consecutive patterns
We prove that the number of permutations avoiding an arbitrary consecutive
pattern of length m is asymptotically largest when the avoided pattern is
12...m, and smallest when the avoided pattern is 12...(m-2)m(m-1). This settles
a conjecture of the author and Noy from 2001, as well as another recent
conjecture of Nakamura. We also show that among non-overlapping patterns of
length m, the pattern 134...m2 is the one for which the number of permutations
avoiding it is asymptotically largest
Asymptotic enumeration of permutations avoiding generalized patterns
Motivated by the recent proof of the Stanley-Wilf conjecture, we study the
asymptotic behavior of the number of permutations avoiding a generalized
pattern. Generalized patterns allow the requirement that some pairs of letters
must be adjacent in an occurrence of the pattern in the permutation, and
consecutive patterns are a particular case of them.
We determine the asymptotic behavior of the number of permutations avoiding a
consecutive pattern, showing that they are an exponentially small proportion of
the total number of permutations. For some other generalized patterns we give
partial results, showing that the number of permutations avoiding them grows
faster than for classical patterns but more slowly than for consecutive
patterns.Comment: 14 pages, 3 figures, to be published in Adv. in Appl. Mat
A bijection between 2-triangulations and pairs of non-crossing Dyck paths
A k-triangulation of a convex polygon is a maximal set of diagonals so that
no k+1 of them mutually cross in their interiors. We present a bijection
between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths
of length 2(n-4). This solves the problem of finding a bijective proof of a
result of Jonsson for the case k=2. We obtain the bijection by constructing
isomorphic generating trees for the sets of 2-triangulations and pairs of
non-crossing Dyck paths.Comment: 17 pages, 12 figure
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