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### The patterns of permutations

Herbert S. Wilf
2002 Discrete Mathematics
Let n; k be positive integers, with k 6 n, and let be a ÿxed permutation of {1; : : : ; k}. 1 We will call the pattern. We will look for the pattern in permutations of n letters.  ...  Then this pattern of k = 3 letters occurs several times in the following permutation , of n = 14 letters (one such occurrence is underlined): = (5 2 9 4 14 10 1 3 6 15 8 11 7 13 12):  ...  permutations. But How can we e ciently list the permutations that avoid other patterns? A lower bound Given a pattern , of k letters. Again, let f(n; ) be the number of n-permutations that avoid .  ...

### On the sub-permutations of pattern avoiding permutations [article]

Filippo Disanto, Thomas Wiehe
2014 arXiv   pre-print
First, given a pattern μ, we study how the avoidance of μ in a permutation π affects the presence of other patterns in the sub-permutations of π.  ...  More precisely, considering patterns of length 3, we solve instances of the following problem: given a class of permutations K and a pattern μ, we ask for the number of permutations π∈ Av_n(μ) whose sub-permutations  ...  Acknowledgement This work was financially supported by grant DFG-SPP1590 from the German Research Foundation to TW.  ...

### Mesh patterns and the expansion of permutation statistics as sums of permutation patterns [article]

Petter Brändén, Anders Claesson
2011 arXiv   pre-print
Intuitively, an occurrence of the mesh pattern p=(π,R) is an occurrence of the permutation pattern π with additional restrictions specified by R on the relative position of the entries of the occurrence  ...  We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns.  ...  So, in terms of mesh patterns, it is the set of permutations that avoid and .  ...

### Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

Petter Brändén, Anders Claesson
2011 Electronic Journal of Combinatorics
Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the  ...  We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns.  ...  So, in terms of mesh patterns, it is the set of permutations that avoid and .  ...

### The feasible regions for consecutive patterns of pattern-avoiding permutations [article]

Jacopo Borga, Raul Penaguiao
2021 arXiv   pre-print
Along the way, we discuss connections of this work with the problem of packing patterns in pattern-avoiding permutations and to the study of local limits for pattern-avoiding permutations.  ...  We study the feasible region for consecutive patterns of pattern-avoiding permutations.  ...  Acknowledgements The authors are very grateful to Valentin Féray and Mathilde Bouvel for some precious discussions during the preparation of this paper.  ...

### The Shape of Random Pattern-Avoiding Permutations [article]

Sam Miner, Igor Pak
2013 arXiv   pre-print
We initiate the study of limit shapes for random permutations avoiding a given pattern.  ...  Specifically, for patterns of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in the permutations.  ...  Acknowledgments: The authors are grateful to Tonći Antunović, Drew Armstrong, Marek Biskup, Stephen DeSalvo, Sergi Elizalde, Sergey Kitaev, Jim Pitman and Richard Stanley, for  ...

### The computational landscape of permutation patterns [article]

Marie-Louise Bruner, Martin Lackner
2014 arXiv   pre-print
In this paper we draw a map of the computational landscape of permutation pattern matching with different types of patterns.  ...  Every type of permutation pattern naturally defines a corresponding computational problem: Given a pattern P and a permutation T (the text), is P contained in T?  ...  Acknowledgements We wish to thank HenningÚlfarsson for drawing our attention to marked mesh, decorated and other types of patterns, Sergey Kitaev for helping us with notational questions and the anonymous  ...

### The Initial Involution Patterns of Permutations

Dongsu Kim, Jang Soo Kim
2007 Electronic Journal of Combinatorics
We also compute the numbers of permutations in $S_n$ with a given $j$-set and prove some properties of them.  ...  We prove a characterization theorem for $j$-sets, give a generating function for the number of different $j$-sets of permutations in $S_n$.  ...  Let π be a permutation and σ be the initial k-pattern of π. For integer i ≤ k, the initial i-pattern of π is equal to the initial i-pattern of σ. Thus J(σ) = J(π)∩[k].  ...

### The Mobius Function of the Permutation Pattern Poset [article]

Einar Steingrimsson, Bridget Eileen Tenner
2010 arXiv   pre-print
We conjecture that for intervals [\sigma,\tau] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of \sigma in \tau.  ...  A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma.  ...  The permutation 246153 also has one occurrence of the pattern 3142, namely the subsequence 4153 .  ...

### The topology of the permutation pattern poset

Peter McNamara, Einar Steingrımsson
2014 Discrete Mathematics & Theoretical Computer Science
International audience The set of all permutations, ordered by pattern containment, forms a poset.  ...  We also present a simplified version of the recursive formula for the Möbius function of decomposable permutations given by Burstein et al.  ...  For example, the permutation 416325 contains two occurrences of the pattern 231, in 463 and 462. The origin of the study of permutation patterns can be traced back a long way.  ...

### The Infinite Limit of Random Permutations Avoiding Patterns of Length Three [article]

Ross G. Pinsky
2018 arXiv   pre-print
For τ∈ S_3, let μ_n^τ denote the uniformly random probability measure on the set of τ-avoiding permutations in S_n.  ...  Extending permutations σ∈ S_n by defining σ_j=j, for j>n, we have S_n⊂ S(N,N^*). For each τ∈ S_3, we study the limiting behavior of the measures {μ_n^τ}_n=1^∞ on S(N,N^*).  ...  Introduction and Statement of Results We recall the definition of pattern avoidance for permutations. Let S n denote the set of permutations of [n] := {1, · · · , n}.  ...

### The shape of random pattern-avoiding permutations

Sam Miner, Igor Pak
We initiate the study of limit shapes for random permutations avoiding a given pattern.  ...  Specifically, for patterns of length 3, we obtain delicate results on the asymptotics of distributions of positions of numbers in the permutations.  ...  We are indebted an anonymous referee for careful reading of the paper, comments and helpful suggestions. The second author was partially supported by the BSF and the NSF.  ...

### On the topology of the permutation pattern poset

Peter R.W. McNamara, Einar Steingrímsson
2015 Journal of combinatorial theory. Series A
The set of all permutations, ordered by pattern containment, forms a poset. This paper presents the first explicit major results on the topology of intervals in this poset.  ...  We also present a simplified version of the recursive formula for the M\"obius function of decomposable permutations given by Burstein et al.  ...  For example, the permutation 416325 contains two occurrences of the pattern 231, in 463 and 462. The origin of the study of permutation patterns can be traced back a long way.  ...

### The Möbius function of the permutation pattern poset

Einar Steingrímsson, Bridget Eileen Tenner
2010 Journal of Combinatorics
We conjecture that for intervals [σ, τ ] consisting of permutations avoiding the pattern 132, the magnitude of the Möbius function is bounded by the number of occurrences of σ in τ .  ...  A permutation τ contains another permutation σ as a pattern if τ has a subsequence whose elements are in the same order with respect to size as the elements in σ.  ...  The permutation 246153 also has one occurrence of the pattern 3142, namely the subsequence 4153 .  ...

### Asymptotics of Pattern Avoidance in the Permutation-Tuple and Klazar Set Partition Settings [article]

Benjamin Gunby
2019 arXiv   pre-print
We consider asymptotics of set partition pattern avoidance in the sense of Klazar.  ...  Several conjectures are proposed, and the related question of asymptotics of parallel (k-tuple) permutation pattern avoidance is considered and solved completely to within an exponential factor, generalizing  ...  Acknowledgements The author would like to thank Ryan Alweiss for inspiring his interest in the problem of the asymptotics of set partition pattern avoidance, Adam Hammett for extremely helpful correspondence  ...
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