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We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2 + 2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not known to be equinumerous. We present a direct bijection between them. The third class is a family of permutations… (More)

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let e k (π) be the number of increasing subsequences of length k + 1 in the permutation π. We prove that any Cata-lan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a… (More)

Motivated by juggling sequences and bubble sort, we examine permutations on the set {1, 2,. .. , n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients. Résumé. Motivés par les "… (More)

Any permutation statistic f : S → C may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: f = Σ τ λ f (τ)τ. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern p = (π, R) is an… (More)

Babson and Steingrímsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1, 2) or (2, 1). For eight of these twelve patterns the answer is… (More)

The napkin problem was first posed by John H. Conway, and written up as a 'toughie' in " Mathematical Puzzles: A Connoisseur's Collection, " by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, n men are to be seated around a circular table. There are n napkins, exactly… (More)

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length n with k holes is a sequence of symbols π = π 1 π 2 · · · π n in which each of the symbols from the set {1, 2,. .. , n − k} appears exactly once, while the remaining k symbols of π are " holes ". 1 We introduce… (More)