## Generalized pattern avoidance

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Venue: | European J. Combin |

Citations: | 72 - 5 self |

### BibTeX

@ARTICLE{Claesson_generalizedpattern,

author = {Anders Claesson},

title = {Generalized pattern avoidance},

journal = {European J. Combin},

year = {},

volume = {22},

pages = {961--971}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Recently, Babson and Steingrímsson have introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We will consider pattern avoidance for such patterns, and give a complete solution for the number of permutations avoiding any single pattern of length three with exactly one adjacent pair of letters. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. We also give some results for the number of permutations avoiding two different patterns. These results relate the permutations in question to Motzkin paths, involutions and non-overlapping partitions. Furthermore, we define a new class of set partitions, called monotone partitions, and show that these partitions are in one-to-one correspondence with non-overlapping partitions. 1.

### Citations

653 | Online Encyclopedia of Integer Sequences
- Sloane
(Show Context)
Citation Context ...avoiding a given set of patterns to other better known combinatorial structures. Here follows a brief description of these structures. Two excellent references on combinatorial structures are [7] and =-=[6]-=-. Set partitions. A partition of a set S is a family, π = {A1, A2, . . . , Ak}, of pairwise disjoint non-empty subsets of S such that S = ∪iAi. We call Ai a block of π. The total number of partitions ... |

267 |
The Art of Computer Programming – Vol.1/Fundamental Algorithms
- Knuth
- 1973
(Show Context)
Citation Context ...ds σ if there is no subsequence in π whose letters are in the same relative order as the letters of σ. For example, π ∈ Sn avoids 132 if there is no 1 ≤ i < j < k ≤ n such that π(i) < π(k) < π(j). In =-=[4]-=- Knuth established that for all σ ∈ S3, the number of permutations in Sn avoiding σ equals the nth Catalan number, Cn = 1 � � 2n 1+n n . One may also consider permutations that are required to avoid s... |

234 |
Enumerative Combinatorics, Vol
- Stanley
- 1986
(Show Context)
Citation Context ...tations avoiding a given set of patterns to other better known combinatorial structures. Here follows a brief description of these structures. Two excellent references on combinatorial structures are =-=[7]-=- and [6]. Set partitions. A partition of a set S is a family, π = {A1, A2, . . . , Ak}, of pairwise disjoint non-empty subsets of S such that S = ∪iAi. We call Ai a block of π. The total number of par... |

204 |
Restricted permutations
- Simion, Schmidt
- 1985
(Show Context)
Citation Context ...hat for all σ ∈ S3, the number of permutations in Sn avoiding σ equals the nth Catalan number, Cn = 1 � � 2n 1+n n . One may also consider permutations that are required to avoid several patterns. In =-=[5]-=- Simion and Schmidt gave a complete solution for permutations avoiding any set of patterns of length three. Even patterns of length greater than three have been considered. For instance, West showed i... |

123 |
The Art of Computer Programming, volume 1
- Knuth
- 1968
(Show Context)
Citation Context ...ds σ if there is no subsequence in π whose letters are in the same relative order as the letters of σ. For example, π ∈ Sn avoids 132 if there is no 1 ≤ i ≤ j ≤ k ≤ n such that π(i) ≤ π(k) ≤ π(j). In =-=[6]-=- Knuth established that for all σ ∈ S3, the number of permutations in Sn avoiding σ equals the nth Catalan number, Cn ( ) = 1 2n 1+n n . One may also consider permutations that are required to avoid s... |

122 | Generalized permutation patterns and a classification of the Mahonian statistics
- Babson, Steingrímsson
(Show Context)
Citation Context ... length greater than three have been considered. For instance, West showed in [8] that permutations avoiding both 3142 and 2413 are enumerated by the Schröder numbers, Sn = �n � � 2n−i i=0 i Cn−i. In =-=[1]-=- Babson and Steingrímsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. The motivation for Babson a... |

105 | Generating trees and the Catalan and Schröder numbers
- West
- 1995
(Show Context)
Citation Context ...imion and Schmidt gave a complete solution for permutations avoiding any set of patterns of length three. Even patterns of length greater than three have been considered. For instance, West showed in =-=[8]-=- that permutations avoiding both 3142 and 2413 are enumerated by the Schröder numbers, Sn = �n � � 2n−i i=0 i Cn−i. In [1] Babson and Steingrímsson introduced generalised permutation patterns that all... |

52 |
Dyck path enumeration
- Deutsch
- 1999
(Show Context)
Citation Context ...π) be the number of left-to-right minima of π. Then � x L(π) = � � � k 2n − k x 2n − k n k . π∈Sn(b ac) k≥0 Proof. Let R(δ) denote the number of return steps in the Dyck path δ. It is well known (see =-=[2]-=-) that the distribution of R over all Dyck paths of length 2n is the distribution we claim that L has over Sn(b ac). . Let γ be a Dyck path of length 2n, and let γ = uαdβ be its first return decomposi... |

18 | Non-overlapping partitions, continued fractions, Bessel functions, and a divergent series
- Flajolet, Schott
- 1990
(Show Context)
Citation Context ... A2, . . . , Ak} is monotone. � We now show that there is a one-to-one correspondence between monotone partitions and non-overlapping partitions. The proof we give is strongly influenced by the paper =-=[3]-=-, in which Flajolet and Schot showed that the ordinary generating function of the Bessel numbers admits a nice continued fraction expansion � B ∗ nx n = n≥0 1 − 1 · x − 1 1 − 2 · x − x 2 x 2 1 − 3 · x... |

11 |
personal communication
- Kitaev
- 1997
(Show Context)
Citation Context ...le 9. The involution π = 826543719 written in standard form is and hence ̂π = 974536218. Porism 10. (9)(7)(45)(36)(2)(18), Sn(a--bc,a--cb) = Sn(a--bc,acb) = Sn(abc,a--cb) = Sn(abc,acb). Proof. Kitaev =-=[5]-=- observed that the dashes in the patterns (a--bc) and (a--cb) are immaterial for the proof of Proposition 8. The result may, however, also be proved directly. For an example of such a proof see the pr... |

4 |
Un modèle combinatoire pour les polynômes de Bessel
- Dulucq, Favreau
- 1990
(Show Context)
Citation Context ...n(x) = n∑ k=0 ( n k )( n − k k ) k! 2 k xn−k . Proof. Let Ik n denote the number of involutions in Sn with k fixed points. Then Porism 11 is equivalently stated as An(x) = ∑ k≥0 I 2k−n n x k . (3) In =-=[3]-=- Dulucq and Favreau showed that the Bessel polynomials are given by yn(x) = ∑ k≥0 I n−k n+k xk . (4)GENERALISED PATTERN AVOIDANCE 9 To prove (i), multiply Equation (4) by (xt) n and sum over n. ∑ yn(... |