We study the difference between the numbers of even and odd permutations in \(\mathfrak{S}_n\) having exactly \(k\) fixed points. We derive a closed formula for this quantity using four complementary approaches: exponential generating functions, a determinant representation, a combinatorial derivation based on inclusion–exclusion on cycle structures, and a factorization via the stabilizer subgroup, through restriction to the complement of the fixed-point set. The resulting expression provides a signed refinement of the classical rencontres numbers and yields a simple polynomial form for the associated signed fixed-point distribution.