permutations(\@data)¶

The permutations of @data are all its reorderings. For example, the permutations of "@data = (1, 2, 3)" are:

    (1, 2, 3)
    (1, 3, 2)
    (2, 1, 3)
    (2, 3, 1)
    (3, 1, 2)
    (3, 2, 1)

The number of permutations of "n" elements is:

    n! = 1,                  if n = 0
    n! = n*(n-1)*...*1,      if n > 0

See some values at <http://www.research.att.com/~njas/sequences/A000142>.

circular_permutations(\@data)¶

The circular permutations of @data are its arrangements around a circle, where only relative order of elements matter, rather than their actual position. Think possible arrangements of people around a circular table for dinner according to whom they have to their right and left, no matter the actual chair they sit on.

For example the circular permutations of "@data = (1, 2, 3, 4)" are:

    (1, 2, 3, 4)
    (1, 2, 4, 3)
    (1, 3, 2, 4)
    (1, 3, 4, 2)
    (1, 4, 2, 3)
    (1, 4, 3, 2)

The number of circular permutations of "n" elements is:

        n! = 1,                      if 0 <= n <= 1
    (n-1)! = (n-1)*(n-2)*...*1,      if n > 1

See a few numbers in a comment of <http://www.research.att.com/~njas/sequences/A000142>.

derangements(\@data)¶

The derangements of @data are those reorderings that have no element in its original place. In jargon those are the permutations of @data with no fixed points. For example, the derangements of "@data = (1, 2, 3)" are:

    (2, 3, 1)
    (3, 1, 2)

The number of derangements of "n" elements is:

    d(n) = 1,                       if n = 0
    d(n) = n*d(n-1) + (-1)**n,      if n > 0

See some values at <http://www.research.att.com/~njas/sequences/A000166>.

complete_permutations(\@data)¶

This is an alias for "derangements", documented above.

variations(\@data, $k)¶

The variations of length $k of @data are all the tuples of length $k consisting of elements of @data. For example, for "@data = (1, 2, 3)" and "$k = 2":

    (1, 2)
    (1, 3)
    (2, 1)
    (2, 3)
    (3, 1)
    (3, 2)

For this to make sense, $k has to be less than or equal to the length of @data.

Note that

    permutations(\@data);

is equivalent to

    variations(\@data, scalar @data);

The number of variations of "n" elements taken in groups of "k" is:

    v(n, k) = 1,                        if k = 0
    v(n, k) = n*(n-1)*...*(n-k+1),      if 0 < k <= n

variations_with_repetition(\@data, $k)¶

The variations with repetition of length $k of @data are all the tuples of length $k consisting of elements of @data, including repetitions. For example, for "@data = (1, 2, 3)" and "$k = 2":

    (1, 1)
    (1, 2)
    (1, 3)
    (2, 1)
    (2, 2)
    (2, 3)
    (3, 1)
    (3, 2)
    (3, 3)

Note that $k can be greater than the length of @data. For example, for "@data = (1, 2)" and "$k = 3":

    (1, 1, 1)
    (1, 1, 2)
    (1, 2, 1)
    (1, 2, 2)
    (2, 1, 1)
    (2, 1, 2)
    (2, 2, 1)
    (2, 2, 2)

The number of variations with repetition of "n" elements taken in groups of "k >= 0" is:

    vr(n, k) = n**k

tuples(\@data, $k)¶

This is an alias for "variations", documented above.

tuples_with_repetition(\@data, $k)¶

This is an alias for "variations_with_repetition", documented above.

combinations(\@data, $k)¶

The combinations of length $k of @data are all the sets of size $k consisting of elements of @data. For example, for "@data = (1, 2, 3, 4)" and "$k = 3":

    (1, 2, 3)
    (1, 2, 4)
    (1, 3, 4)
    (2, 3, 4)

For this to make sense, $k has to be less than or equal to the length of @data.

The number of combinations of "n" elements taken in groups of "0 <= k <= n" is:

    n choose k = n!/(k!*(n-k)!)

combinations_with_repetition(\@data, $k);¶

The combinations of length $k of an array @data are all the bags of size $k consisting of elements of @data, with repetitions. For example, for "@data = (1, 2, 3)" and "$k = 2":

    (1, 1)
    (1, 2)
    (1, 3)
    (2, 2)
    (2, 3)
    (3, 3)

Note that $k can be greater than the length of @data. For example, for "@data = (1, 2, 3)" and "$k = 4":

    (1, 1, 1, 1)
    (1, 1, 1, 2)
    (1, 1, 1, 3)
    (1, 1, 2, 2)
    (1, 1, 2, 3)
    (1, 1, 3, 3)
    (1, 2, 2, 2)
    (1, 2, 2, 3)
    (1, 2, 3, 3)
    (1, 3, 3, 3)
    (2, 2, 2, 2)
    (2, 2, 2, 3)
    (2, 2, 3, 3)
    (2, 3, 3, 3)
    (3, 3, 3, 3)

The number of combinations with repetition of "n" elements taken in groups of "k >= 0" is:

    n+k-1 over k = (n+k-1)!/(k!*(n-1)!)

partitions(\@data[, $k])¶

A partition of @data is a division of @data in separate pieces. Technically that's a set of subsets of @data which are non-empty, disjoint, and whose union is @data. For example, the partitions of "@data = (1, 2, 3)" are:

    ((1, 2, 3))
    ((1, 2), (3))
    ((1, 3), (2))
    ((1), (2, 3))
    ((1), (2), (3))

This subroutine returns in consequence tuples of tuples. The top-level tuple (an arrayref) represents the partition itself, whose elements are tuples (arrayrefs) in turn, each one representing a subset of @data.

The number of partitions of a set of "n" elements are known as Bell numbers, and satisfy the recursion:

    B(0) = 1
    B(n+1) = (n over 0)B(0) + (n over 1)B(1) + ... + (n over n)B(n)

See some values at <http://www.research.att.com/~njas/sequences/A000110>.

If you pass the optional parameter $k, the subroutine generates only partitions of size $k. This uses an specific algorithm for partitions of known size, which is more efficient than generating all partitions and filtering them by size.

Note that in that case the subsets themselves may have several sizes, it is the number of elements of the partition which is $k. For instance if @data has 5 elements there are partitions of size 2 that consist of a subset of size 2 and its complement of size 3; and partitions of size 2 that consist of a subset of size 1 and its complement of size 4. In both cases the partitions have the same size, they have two elements.

The number of partitions of size "k" of a set of "n" elements are known as Stirling numbers of the second kind, and satisfy the recursion:

    S(0, 0) = 1
    S(n, 0) = 0 if n > 0
    S(n, 1) = S(n, n) = 1
    S(n, k) = S(n-1, k-1) + kS(n-1, k)

subsets(\@data[, $k])¶

This subroutine iterates over the subsets of data, which is assumed to represent a set. If you pass the optional parameter $k the iteration runs over subsets of data of size $k.

The number of subsets of a set of "n" elements is

  2**n

See some values at <http://www.research.att.com/~njas/sequences/A000079>.

Warnings¶

The following warnings may be issued:

Useless use of %s in void context

A subroutine was called in void context.

Parameter k is negative

A subroutine was called with a negative k.

Parameter k is greater than the size of data

A subroutine that does not generate tuples with repetitions was called with a k greater than the size of data.

Errors¶

The following errors may be thrown:

Missing parameter data

A subroutine was called with no parameters.

Missing parameter k

A subroutine that requires a second parameter k was called without one.

Parameter data is not an arrayref

The first parameter is not an arrayref (tested with "reftype()" from Scalar::Util.)