# Pattern avoidance in partial permutations

Claesson, A. and Jelinek, V. and Jelinkova, E. and Kitaev, S.
(2011)
*Pattern avoidance in partial permutations.*
The Electronic Journal of Combinatorics, 18 (1).
ISSN 1077-8926

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## 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.

#### ORCID iDs

Claesson, A. ORCID: https://orcid.org/0000-0001-5797-8673, Jelinek, V., Jelinkova, E. and Kitaev, S. ORCID: https://orcid.org/0000-0003-3324-1647;Item type: Article ID code: 40421 Dates: DateEvent2011PublishedKeywords: Wilf-equivalence, generating function, pattern avoidance, partial permutation, Stanley-Wilf conjecture, bijection, partial words, fillings, Baxter permutation, Electronic computers. Computer science, Computational Theory and Mathematics, Geometry and Topology, Theoretical Computer Science Subjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 17 Jul 2012 11:54 Last modified: 21 Mar 2021 01:26 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/40421