# Pattern avoidance in partial permutations

A. Claesson, V. Jelinek, E. Jelinkova, S. Kitaev

Research output: Contribution to journalArticle

5 Citations (Scopus)

### Abstract

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 $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the set {1,2,...,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 >= k >= 1.
Original language English 41 The Electronic Journal of Combinatorics 18 1 Published - 2011

### Fingerprint

Pattern Avoidance
Equivalence classes
Permutation
Partial
Pi
Partial Words
Equivalence class
Equivalence

### Keywords

• Wilf-equivalence
• generating function
• pattern avoidance
• partial permutation
• Stanley-Wilf conjecture
• bijection
• partial words
• fillings
• Baxter permutation

### Cite this

Claesson, A. ; Jelinek, V. ; Jelinkova, E. ; Kitaev, S. / Pattern avoidance in partial permutations. In: The Electronic Journal of Combinatorics. 2011 ; Vol. 18, No. 1.
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title = "Pattern avoidance in partial permutations",
abstract = "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 $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the set {1,2,...,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 >= k >= 1.",
keywords = "Wilf-equivalence, generating function, pattern avoidance, partial permutation, Stanley-Wilf conjecture, bijection, partial words, fillings, Baxter permutation",
author = "A. Claesson and V. Jelinek and E. Jelinkova and S. Kitaev",
year = "2011",
language = "English",
volume = "18",
journal = "The Electronic Journal of Combinatorics",
issn = "1077-8926",
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Claesson, A, Jelinek, V, Jelinkova, E & Kitaev, S 2011, 'Pattern avoidance in partial permutations', The Electronic Journal of Combinatorics, vol. 18, no. 1.

Pattern avoidance in partial permutations. / Claesson, A.; Jelinek, V.; Jelinkova, E.; Kitaev, S.

In: The Electronic Journal of Combinatorics, Vol. 18, No. 1, 2011.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Pattern avoidance in partial permutations

AU - Claesson, A.

AU - Jelinek, V.

AU - Kitaev, S.

PY - 2011

Y1 - 2011

N2 - 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 $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the set {1,2,...,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 >= k >= 1.

AB - 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 $\pi = \pi_1\pi_2 ... \pi_n$ in which each of the symbols from the set {1,2,...,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 >= k >= 1.

KW - Wilf-equivalence

KW - generating function

KW - pattern avoidance

KW - partial permutation

KW - Stanley-Wilf conjecture

KW - bijection

KW - partial words

KW - fillings

KW - Baxter permutation

UR - http://arxiv.org/abs/1005.2216

UR - https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/partial-permutations.pdf

UR - http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p25/pdf

M3 - Article

VL - 18

JO - The Electronic Journal of Combinatorics

JF - The Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -