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The patterns of permutations

2002
*
Discrete Mathematics
*

Let n; k be positive integers, with k 6 n, and let be a ÿxed

doi:10.1016/s0012-365x(02)00515-0
fatcat:4mfhdis5abeeffwcw6jvxccfgm
*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 . ...##
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On the sub-permutations of pattern avoiding permutations
[article]

2014
*
arXiv
*
pre-print

First, given a

arXiv:1210.6908v4
fatcat:uk7dxt7zgratnckhqnmy2po6ie
*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. ...##
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Mesh patterns and the expansion of permutation statistics as sums of permutation patterns
[article]

2011
*
arXiv
*
pre-print

Intuitively, an occurrence

arXiv:1102.4226v2
fatcat:xgzbvobda5d4lpkgjz5geihuou
*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

2011
*
Electronic Journal of Combinatorics
*

Intuitively, an occurrence

doi:10.37236/2001
fatcat:skswqrflhfeadmt36wp4crlvse
*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]

2021
*
arXiv
*
pre-print

Along

arXiv:2010.06273v3
fatcat:63gvkedw5rhpvgmrg4w2myj4qq
*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]

2013
*
arXiv
*
pre-print

We initiate

arXiv:1303.7313v3
fatcat:a4yr5xdvuzg25lyzatlpez2uia
*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]

2014
*
arXiv
*
pre-print

In this paper we draw a map

arXiv:1301.0340v2
fatcat:of4huwt4dna2vd2gusyvpxpm5e
*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

2007
*
Electronic Journal of Combinatorics
*

We also compute

doi:10.37236/921
fatcat:cwzxtzpfjnfrbbynctbq3fgiju
*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]. ...##
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The Mobius Function of the Permutation Pattern Poset
[article]

2010
*
arXiv
*
pre-print

We conjecture that for intervals [\sigma,\tau] consisting

arXiv:0902.4011v3
fatcat:4egskjl5gfgoxmx4zfvv5lp46q
*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

2014
*
Discrete Mathematics & Theoretical Computer Science
*

International audience

doi:10.46298/dmtcs.2397
fatcat:sxxorlrixfhy7c4hufompynwya
*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]

2018
*
arXiv
*
pre-print

For τ∈ S_3, let μ_n^τ denote

arXiv:1806.07669v3
fatcat:w2jpl7gbsrgb5pemziaf6lyfuu
*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

2014
*
Advances in Applied Mathematics
*

We initiate

doi:10.1016/j.aam.2013.12.004
fatcat:ihkzi5jdo5hejcebnje67hn7he
*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

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

2010
*
Journal of Combinatorics
*

We conjecture that for intervals [σ, τ ] consisting

doi:10.4310/joc.2010.v1.n1.a3
fatcat:t4ihkukbjbac7c7t3ljsbx7nli
*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]

2019
*
arXiv
*
pre-print

We consider asymptotics

arXiv:1609.06023v3
fatcat:4puadvapvbeztanwzcx7x4cawu
*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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