Definition of the permanent

Let M be an n×n matrix over a field (or even a commutative ring) 𝖨𝖥, and Mij its entries (with i ranging over a set I and j over a set J, each of n elements).

The permanentMathworldPlanetmath of M, denoted by perM, is given by


where S is the set of all n! bijections π:IJ. Usually I and J are identified with each other (and with the set of the first n natural numbersMathworldPlanetmath, traditionally skipping 0) so that S consists of the elements of the symmetric groupMathworldPlanetmathPlanetmath Sn, the group of permutationsMathworldPlanetmath of n objects.

In words: we want productsPlanetmathPlanetmath of each time n matrix elements chosen such that there’s one from each row i and also one from each column j. There are n! ways to pick those elements (for any permutation of the column indices relative to the row indices take the elements that end up in diagonal position). The permanent is the sum of those n! products. E.g. (n=3)

@ O O O @ O O O @ @ O O O @ O O O @
O @ O + O O @ + @ O O + O O @ + @ O O + O @ O
O O @ @ O O O @ O O @ O O O @ @ O O

Comparison with the determinant

It is closely related to one of the ways to define the determinantMathworldPlanetmath,


where SI is the symmetry group of I (isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to Sn); the sign is + for even permutationsMathworldPlanetmath and - for odd ones. E.g. (n=3)

@ O O O @ O O O @ @ O O O @ O O O @
O @ O + O O @ + @ O O - O O @ - @ O O - O @ O
O O @ @ O O O @ O O @ O O O @ @ O O

Note two important differencesPlanetmathPlanetmath though.

  • While the determinant enjoys the property that (detA)(detB)=det(AB), the permanent has no such nice arithmetic properties.

  • In the definition of the determinant, we must have I=J i.e. we must have a particular way in mind of matching rows to columns. A matrix to transform from one basis to another would be an example where this match is arbitrary. Different conventions which row goes with which column give two different values (one -1 times the other) for the determinant. By contrast, the permanent is well defined for any I and J.

In the representation theory of groups (where the field 𝖨𝖥 is ıı𝖢) determinant and permanent are special cases of the immanent (there is an immanent for every character of Sn).

Some properties of the permanent

These follow immediately from the definition:

  • The permanent is “multilinear” in the rows and columns (i.e. linear in every one of them).

  • It is “homogeneous of degree n”, i.e. per(kA)=knper(A), where k is a scalar (i.e. element of 𝖨𝖥).

  • When P and Q are permutation matricesMathworldPlanetmath, per(PAQ)=per(PA)=per(AQ)=per(A).

  • per(A)=per(A).

  • per(A  0 0D)=per(A)per(D).

  • If M only has nonnegative entries, then


    Equality is attained

    • (with value 0) when the products (of entries, one from each row and from each column) that contribute to det and per are all zero. This happens whenever a whole column or row is zero, but also for instance in

    • when the permutations used are all even (which implies only one is used). This happens in a diagonal matrixMathworldPlanetmath, but also for instance in


Permanents find application in combinatorics, e.g. in enumerating Latin squaresMathworldPlanetmath.

See also Van der Waerden’s permanent conjecture (now a theorem) for doubly stochastic matrices (where 𝖨𝖥 is 𝖨𝖱).

Note: some authors define something for non-square matrices they also call “permanent”.

Title permanent
Canonical name Permanent
Date of creation 2013-03-22 14:05:39
Last modified on 2013-03-22 14:05:39
Owner marijke (8873)
Last modified by marijke (8873)
Numerical id 12
Author marijke (8873)
Entry type Definition
Classification msc 15A15
Related topic Immanent
Related topic Determinant2