q-Permanent

The -permanent of an matrix is the polynomial

(1)

where is the symmetric group and counts the permutation inversions of . It specializes to the determinant at and the permanent at (Bapat and Lal 1994).

The conjectured monotonicity on for real symmetric positive definite matrices must be distinguished from a proposed extension beyond 1. da Fonseca and Goyal (2026) refuted the extension, showing that every real admits a real symmetric positive definite matrix with . The smallest order at which monotonicity can fail for some is 4. For example,

(2)

is positive definite and satisfies

(3)

The leading principal minors are 52, 1863, 7772, and 6432. There are also positive-definite examples with for some , and examples of order whose derivative is negative at a prescribed , as decreases to 1. Here denotes big-O notation. These counterexamples do not settle the real symmetric monotonicity conjecture on . da Fonseca and Goyal (2026) credit GPT-6 with finding the initial counterexamples.