(mathematics, statistical mechanics:) The permanent is a function of square matrices

```   a_11 ... a_1n
a_21 ... a_2n
...  ... ...
a_n1 ... a_nn
```
defined as the sum over all permutations p of the integers 1...n of the product a_{1,p(1)} * ... * a_{n,p(n)}:
Perm(aij) = p∈Sn i=1n ai,p(i).

Note that it looks a lot like the determinant, except that there the signs of the sum alternate with the permutation. However, while good polynomial algorithms are known for the determinant, no easy way to calculate the permanent is known.

Per"ma*nent (?), a. [L. permanens, -entis, p.pr. of permanere to stay or remain to the end, to last; per + manere to remain: cf. F. permanent. See Per-, and Mansion.]

Continuing in the same state, or without any change that destroys form or character; remaining unaltered or unremoved; abiding; durable; fixed; stable; lasting; as, a permanent impression.

Eternity stands permanent and fixed. Dryden.

Permanent gases Chem. & Physics, hydrogen, oxygen, nitrogen, and carbon monoxide; -- also called incondensible ∨ incoercible gases, before their liquefaction in 1877.<-- archaic --> -- Permanent way, the roadbed and superstructure of a finished railway; -- so called in distinction from the contractor's temporary way. -- Permanent white Chem., barium sulphate (heavy spar), used as a white pigment or paint, in distinction from white lead, which tarnishes and darkens from the formation of the sulphide.

Syn. -- Lasting; durable; constant. See Lasting.