A permutation vector
maps a set of \(n\) objects \(\{O_1, O_2, ..., O_n\}\) onto itself.
Ordering Representation:
In seriation we represent a permutation \(\pi\)
as a vector which lists the objects' indices in their permuted order. This can
be seen as replacing the object in position \(i\) with the object
in position \(\pi(i)\).
For example, the permutation vector \(\langle3, 1, 2\rangle\) indicates that in
first position is the object with index 3 then the object with index 1 and finally
the object with index 2. This representation is often called a (re)arrangement or ordering.
The ordering can be extracted from a permutation vector object
via get_order(). Such an ordering can be directly used
to subset the list of original objects with "[" to apply the permutation.
Rank Representation:
An alternative way to specify a permutation is via a list of the ranks
of the objects after permutation. This representation is often called
a map or substitution. Ranks can be extracted from a permutation vector using get_rank().
Permutation Matrix:
Another popular representation is a permutation matrix which performs
permutations using matrix multiplication. A permutation matrix can be obtained
using get_permutation_matrix().
ser_permutation_vector objects are usually packed into
a ser_permutation object
which is a collection (a list) of \(k\) permutation vectors for \(k\)-mode data.
The constructor ser_permutation_vector()
checks if the permutation vector is valid
(i.e. if all integers occur exactly once).