Editing Permanent (mathematics) (section)

== Definition ==
The permanent of an {{math|''n''×''n''}} matrix {{math|1=''A'' = (''a''<sub>''i,j''</sub>)}} is defined as

<math display="block"> \operatorname{perm}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.</math>

The sum here extends over all elements σ of the [[symmetric group]] ''S''<sub>''n''</sub>; i.e. over all [[permutation]]s of the numbers 1, 2, ..., ''n''.

For example,

<math display="block">\operatorname{perm}\begin{pmatrix}a&b \\ c&d\end{pmatrix}=ad+bc,</math>

and

<math display="block">\operatorname{perm}\begin{pmatrix}a&b&c \\ d&e&f \\ g&h&i \end{pmatrix}=aei + bfg + cdh + ceg + bdi + afh.</math>

The definition of the permanent of ''A'' differs from that of the [[determinant]] of ''A'' in that the [[signature (permutation)|signatures]] of the permutations are not taken into account.

The permanent of a matrix A is denoted per ''A'', perm ''A'', or Per ''A'', sometimes with parentheses around the argument. Minc uses Per(''A'') for the permanent of rectangular matrices, and per(''A'') when ''A'' is a square matrix.<ref>{{harvtxt|Minc|1978}}</ref> Muir and Metzler use the notation <math>\overset{+}{|}\quad \overset{+}{|}</math>.<ref>{{harvtxt|Muir|Metzler|1960}}</ref>

The word, ''permanent'', originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function,<ref>{{Citation| last=Cauchy | first=A. L.| title=Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment. |url=https://gallica.bnf.fr/ark:/12148/bpt6k90193x/f97 |journal=Journal de l'École Polytechnique |volume=10 |pages=91–169 |year=1815}}</ref> and was used by Muir and Metzler<ref>{{harvtxt|Muir|Metzler|1960}}</ref> in the modern, more specific, sense.<ref>{{harvnb|van Lint|Wilson|2001|loc=p. 108}}</ref>