  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.

Combinatorics