Pattern avoidance in partial permutations

TL;DR

This paper introduces the concept of partial permutations with holes, extending pattern-avoidance theory from permutations to partial permutations, and provides enumeration results for pattern-avoiding partial permutations.

Contribution

It extends Wilf equivalence results to partial permutations with holes and establishes connections with Baxter permutations.

Findings

Most Wilf equivalence results extend to partial permutations with holes.

Baxter permutations relate to a Wilf-type equivalence class with partial permutations.

Enumerates pattern-avoiding partial permutations for patterns of length up to four.

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.