Pattern avoidance in partial permutations

Details Pattern avoidance in partial permutations Pattern avoidance in partial permutations

2010-05-12

arxiv.org/abs/1005.2216v1

The Electronic Journal of Combinatorics 18(1), #P25 (2011)

Mathematics

Combinatorics

34 pages, 17 figures. Submitted to the Electronic Journal of Combinatorics. An extended abstract will appear in proceedings of

Scientific paper

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.

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