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{\bf Anders Claesson, V\'{\i}t Jel\'{\i}nek, Eva Jel\'{\i}nkov\'a and Sergey Kitaev}
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{\bf Pattern Avoidance in Partial Permutations}
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Motivated by the concept of partial words, we introduce an analogous
concept of partial permutations. A \emph{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 $n\ge k\ge 1$.

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