Examples for

Permutations

The permutation is an important operation in combinatorics and in other areas of mathematics. To permute a list is to rearrange its elements. To count the permutations of a list is to count the number of unique rearrangements of the list. Wolfram|Alpha is useful for counting, generating and doing algebra with permutations.
Algebra of Permutations

Perform calculations using permutations and analyze their properties.

Describe a permutation:

permutation (1 3 5)(2 4)(6 7 8)

Do algebra with permutations:

perm (1 2 3 4)^3(1 2 3)^-1

Compute properties of a permutation:

signature of permutation (1 2 3 4 5)(6 7 8 9)
matrix of the permutation (3 1 2 5)
factoradic form of the permutation (3 1 2 5 4)
index of (1 4 5)(2 3 6 7)
order of permutation (2 1 3 4)^-1
lexicographic rank of (1 2 6)^2
More examples
Random Permutations

Generate random permutations of a specified number of elements.

Generate a random permutation:

random permutation on 15 elements
More examples
Permutations of a Set

Permute different types of data, such as sets with no repeat elements and lists that allow repetition.

Compute permutations of a set:

permutations of {a, b, c, d}

Compute distinct permutations of a list:

distinct permutations of {1, 2, 2, 3, 3, 3}
More examples
Counting Permutations

Count how many rearrangements of a set or list are unique. Use different list sizes and permutation sizes.

Count permutations:

number of permutations of 23 elements

Specify a permutation size:

number of 6-permutations of 15 objects

Count derangements:

derangements on 12 elements
More examples