scholarly journals Pattern Avoidance in Partial Permutations

10.37236/512 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Wilf Equivalence ◽  
Partial Words

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 $\pi=\pi_1\pi_2\dotsb\pi_n$ in which each of the symbols from the set $\{1,2,\dotsc,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ 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 $n\ge k\ge 1$.

scholarly journals Pattern avoidance in partial permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Nous Montrons ◽  
International Audience ◽  
Wilf Equivalence ◽  
Partial Words

International audience Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ 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 $n \geq k \geq 1$. Nous introduisons un concept de permutations partielles. $\textit{Une permutation partielle de longueur n avec k trous}$ est une suite finie de symboles $\pi = \pi_1 \pi_2 \cdots \pi_n$ dans laquelle chaque nombre de l'ensemble $\{1,2,\ldots,n-k\}$ apparaît précisément une fois, tandis que les $k$ autres symboles de $\pi$ sont des "trous''. Nous introduisons l'étude des permutations partielles à motifs exclus et nous montrons que la plupart des résultats sur l'équivalence de Wilf peuvent être généralisés aux permutations partielles avec un nombre arbitraire de trous. De plus, nous montrons que les permutations de Baxter d'une longueur donnée $k$ forment une classe d'équivalence du type Wilf par rapport aux permutations partielles avec $(k-2)$ trous. Enfin, nous présentons l'énumération des permutations partielles de longueur $n$ avec $k$ trous qui évitent un motif de longueur $\ell \leq 4$, pour chaque $n \geq k \geq 1$.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Abelian pattern avoidance in partial words

2014 ◽  
Vol 48 (3) ◽  
pp. 315-339 ◽  
Author(s):  
F. Blanchet-Sadri ◽  
Benjamin De Winkle ◽  
Sean Simmons
Keyword(s):  
Pattern Avoidance ◽  
Partial Words

scholarly journals Pattern Avoidance by Even Permutations

10.37236/2024 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Andrew Baxter ◽  
Aaron D. Jaggard
Keyword(s):  
Symmetric Group ◽  
Pattern Avoidance ◽  
Alternating Group ◽  
Wilf Equivalence

We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering even-Wilf-equivalence analogues, we prove that other Wilf-equivalence results continue to hold in the even-Wilf-equivalence setting. In particular, we prove that $t(t-1)\cdots 321$ and $(t-1)(t-2)\cdots 21t$ are even-shape-Wilf-equivalent for odd $t$, paralleling a result (which held for all $t$) of Backelin, West, and Xin for shape-Wilf-equivalence. This allows us to classify the symmetric group $\mathcal{S}_{4}$, and to partially classify $\mathcal{S}_{5}$ and $\mathcal{S}_{6}$, according to even-Wilf-equivalence. As with transition to involution-Wilf-equivalence, some—but not all—of the classical Wilf-equivalence results are preserved when we make the transition to even-Wilf-equivalence.

scholarly journals Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

10.37236/3246 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Nihal Gowravaram ◽  
Ravi Jagadeesan
Keyword(s):  
Pattern Avoidance ◽  
General Context ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Alternating Permutations ◽  
Equivalence Type ◽  
Shape Equivalence ◽  
Pattern Avoiding Permutations ◽  
Wilf Equivalence

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

scholarly journals On Partially Ordered Patterns of Length 4 and 5 in Permutations

10.37236/8605 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Alice L. L. Gao ◽  
Sergey Kitaev
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Relative Order ◽  
Permutation Patterns ◽  
Integer Sequences ◽  
Arbitrary Length ◽  
Partially Ordered ◽  
Long Line ◽  
Online Encyclopedia ◽  
Wilf Equivalence

Partially ordered patterns (POPs) generalize the notion of classical patterns studied widely in the literature in the context of permutations, words, compositions and partitions. In an occurrence of a POP, the relative order of some of the elements is not important. Thus, any POP of length $k$ is defined by a partially ordered set on $k$ elements, and classical patterns correspond to $k$-element chains. The notion of a POP provides  a convenient language to deal with larger sets of permutation patterns. This paper contributes to a long line of research on classical permutation patterns of length 4 and 5, and beyond, by conducting a systematic search of connections between sequences in the Online Encyclopedia of Integer Sequences (OEIS) and permutations avoiding POPs of length 4 and 5. As the result, we (i) obtain  13 new enumerative results for classical patterns of length 4 and 5, and a number of results for patterns of arbitrary length, (ii) collect under one roof many sporadic results in the literature related to avoidance of patterns of length 4 and 5, and (iii) conjecture 6 connections to the OEIS. Among the most intriguing bijective questions we state, 7 are related to explaining Wilf-equivalence of various sets of patterns, e.g. 5 or 8 patterns of length 4, and 2 or 6 patterns of length 5.

scholarly journals Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

10.37236/147 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Parking Functions ◽  
Arbitrary Size ◽  
Wilf Equivalence ◽  
Full Classification

In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.

scholarly journals Pattern avoidance in partial words over a ternary alphabet

2015 ◽  
Vol 69 (1) ◽  
Author(s):  
Adam Gągol
Keyword(s):  
Pattern Avoidance ◽  
Partial Words

AbstractBlanched-Sadri and Woodhouse in 2013 have proven the conjecture of Cassaigne, stating that any pattern with m distinct variables and of length at least 2

ALGORITHMIC COMBINATORICS ON PARTIAL WORDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1189-1206 ◽  
Author(s):  
F. BLANCHET-SADRI
Keyword(s):  
Molecular Biology ◽  
Data Compression ◽  
Pattern Avoidance ◽  
Finite Alphabet ◽  
Open Problems ◽  
The Past ◽  
Partial Word ◽  
De Bruijn ◽  
Subword Complexity ◽  
Partial Words

Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

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