Permanent

An analog of a Determinant where all the signs in the expansion by Minors are taken as Positive. The permanent of a Matrix A is the coefficient of $x_1\cdots x_n$ in

$$\prod_{i=1}^n (a_{i1}x_1+a_{i2}x_2+\ldots+a_{in}x_n)$$

(Vardi 1991). Another equation is the Ryser Formula

$$\mathop{\rm perm}(a_{ij})=(-1)^n\sum_{s\subseteq \{1, \ldots, n\}} (-1)^{\vert s\vert} \prod_{i=1}^n \sum_{j\in s} a_{ij},$$

where the Sum is over all Subsets of $\{1, \ldots, n\}$, and $\vert s\vert$ is the number of elements in $s$ (Vardi 1991).

If ${\hbox{\sf M}}$ is a Unitary Matrix, then

$$\vert\mathop{\rm perm}({\hbox{\sf M}})\vert\leq 1$$

(Minc 1978, p. 25; Vardi 1991).

See also Determinant, Frobenius-König Theorem, Immanant, Ryser Formula, Schur Matrix

References

Borovskikh, Y. V. and Korolyuk, V. S. Random Permanents. Philadelphia, PA: Coronet Books, 1994.

Minc, H. Permanents. Reading, MA: Addison-Wesley, 1978.

Vardi, I. ``Permanents.'' §6.1 in Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 108 and 110-112, 1991.