The permanent of a square matrix is defined similar to its determinant, without signs on the products of entries. I will introduce a notion of permanent for elements of the group ring of an infinite amenable group with non-negative real coefficients, and compare it with the Fuglede-Kadison determinant. I will also discuss why the permanent computes the topological pressure of the subshifts associated to restricted permutations. This is based on a joint work with Klaus Schmidt.