Workshop permutation patterns 2005 List of Abstracts (original) (raw)

Pattern avoidance in flattened permutations

To flatten a permutation expressed as a product of disjoint cycles, we mean to form another permutation by erasing the parentheses which enclose the cycles of the original. This clearly depends on how the cycles are listed. For permutations written in the standard cycle form-cycles arranged in increasing order of their first entries, with the smallest element first in each cycle-we count the permutations of [n] whose flattening avoids any subset of S3. Among the sequences that arise are central binomial coefficients, Schröder numbers, and relatives of the Fibonacci numbers. In some instances, we provide combinatorial arguments of the result, while in others, our approach is more algebraic. In a couple of the cases, we define an explicit bijection between the subset of Sn in question and a restricted set of lattice paths. In another, to establish the result, we make use of the kernel method to solve a functional equation arising once a certain parameter has been considered.

Enumeration Schemes for Permutations Avoiding Barred Patterns

We give the first comprehensive collection of enumeration results for permutations that avoid barred patterns of length 4. We then use the method of prefix enumeration schemes to find recurrences counting permutations that avoid a barred pattern of length > 4 or a set of barred patterns. * The author thanks an anonymous referee for several useful suggestions that simplified the organization of this paper. the electronic journal of combinatorics 17 (2010), #R29

Consecutive patterns in restricted permutations and involutions

2019

It is well-known that the set mathbfIn\mathbf I_nmathbfIn of involutions of the symmetric group mathbfSn\mathbf S_nmathbfSn corresponds bijectively - by the Foata map FFF - to the set of nnn-permutations that avoid the two vincular patterns underline123,\underline{123},underline123, underline132.\underline{132}.underline132. We consider a bijection Gamma\GammaGamma from the set mathbfSn\mathbf S_nmathbfSn to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to mathbfSn(1underline23,1underline32).\mathbf S_n(1\underline{23},1\underline{32}).mathbfSn(1underline23,1underline32). In particular, we show that the set mathbfSn(underline123,132)\mathbf S_n(\underline{123},{132})mathbfSn(underline123,132) of permutations that avoids the consecutive pattern underline123\underline{123}underline123 and the classical pattern 132132132 corresponds via Gamma\GammaGamma to the set of Motzkin paths, while its image under FFF is the set of restricted involutions mathbfIn(3412).\mathbf I_n(3412).mathbfIn(3412). We exploit these results to determine the joint distribution of the statistics des and inv over mathbfSn(underline123,132)\mathbf S_n(\underline{123},{132})mathbfSn(underline123,132) and over mathbfIn(3412).\mathbf I_n(3412).mathbfIn(3412). Moreover, we determine the d...

Pattern avoidance in partial permutations

The electronic journal of combinatorics

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 pi=pi1pi2...pi_n\pi = \pi_1\pi_2 ... \pi_npi=pi_1pi2...pin in which each of the symbols from the set {1,2,...,n-k} appears exactly once, while the remaining k symbols of pi\pipi are "holes". We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length k correspond to a Wilf-type equivalence class with respect to partial permutations with (k-2) holes. Lastly, we enumerate the partial permutations of length n with k holes avoiding a given pattern of length at most four, for each n >= k >= 1.

The Enumeration of Maximally Clustered Permutations

Annals of Combinatorics, 2010

The maximally clustered permutations are characterized by avoiding the classical permutation patterns {3421, 4312, 4321}. This class contains the freely braided permutations and the fully commutative permutations. In this work, we show that the generating functions for certain fully commutative pattern classes can be transformed to give generating functions for the corresponding freely braided and maximally clustered pattern classes. Moreover, this transformation of generating functions is rational. As a result, we obtain enumerative formulas for the pattern classes mentioned above as well as the corresponding hexagon-avoiding pattern classes where the hexagon-avoiding permutations are characterized by avoiding {46718235, 46781235, 56718234, 56781234}.

Permutations Containing and Avoiding Certain Patterns

Formal Power Series and Algebraic Combinatorics, 2000

Let T m k = {σ ∈ S k | σ 1 = m}. We prove that the number of permutations which avoid all patterns in T m k equals (k − 2)!(k − 1) n+1−k for k ≤ n. We then prove that for any τ ∈ T 1 k (or any τ ∈ T k k), the number of permutations which avoid all patterns in T 1 k (or in T k k) except for τ and contain τ exactly once equals (n + 1 − k)(k − 1) n−k for k ≤ n. Finally, for any τ ∈ T m k , 2 ≤ m ≤ k − 1, this number equals (k − 1) n−k for k ≤ n. These results generalize recent results due to Robertson concerning permutations avoiding 123-pattern and containing 132-pattern exactly once.

Geometric grid classes of permutations

Transactions of the American Mathematical Society, 2013

A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ± 1 \pm 1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.

Permutations avoiding 1324 and patterns in Łukasiewicz paths

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.