Partial permutation

Combinatorial enumeration

The number of partial permutations on n items, for n = 0, 1, 2, ..., is given by the integer sequence

1, 2, 7, 34, 209, 1546, 13327, 130922, 1441729, 17572114, 234662231, ... (sequence A002720 in the OEIS)

where the nth item in the sequence is given by the summation formula

${\displaystyle \sum _{i=0}^{n}i!{(\genfrac{}{}{0ex}{}{n}{i})}^2}$

in which the ith term counts the number of partial permutations with support of size i, that is, the number of partial permutations with i non-hole entries. Alternatively, it can be computed by a recurrence relation

${\displaystyle P(n)=2nP(n-1)-(n-1{)}^{2}P(n-2).}$

This is determined as follows:

1. ${\displaystyle P(n-1)}$ partial permutations where the final elements of each set are omitted:
2. ${\displaystyle P(n-1)}$ partial permutations where the final elements of each set map to each other.
3. ${\displaystyle (n-1)P(n-1)}$ partial permutations where the final element of the first set is included, but does not map to the final element of the second set
4. ${\displaystyle (n-1)P(n-1)}$ partial permutations where the final element of the second set is included, but does not map to the final element of the first set
5. ${\displaystyle -(n-1{)}^{2}P(n-2)}$, the partial permutations included in both counts 3 and 4, those permutations where the final elements of both sets are included, but do not map to each other.

Restricted partial permutations

Some authors restrict partial permutations so that either the domain or the range of the bijection is forced to consist of the first k items in the set of n items being permuted, for some k. In the former case, a partial permutation of length k from an n-set is just a sequence of k terms from the n-set without repetition. (In elementary combinatorics, these objects are sometimes confusingly called "k-permutations" of the n-set.)