arXiv:2611.14694v1 Announce Type: new Abstract: The $q$-permanent $P_q$ of an $n\times n$ matrix weights the term of each permutation $\sigma$ by $q^{\inv(\sigma)}$, where $\inv(\sigma)$ is the number of inversions of $\sigma$. It interpolates between the determinant ($q=-1$) and the permanent ($q=1$). Extending a conjecture of Bapat on $[-1,1]$, it was conjectured in 2010 that the $q$-permanent of every non-diagonal real symmetric positive definite matrix is strictly increasing on an interval $(\varepsilon,+\infty)$ with $\varepsilon<-1$. In this note, we disprove this conjecture. An explicit real symmetric positive definite integer matrix of order five, whose $q$-permanent has only nine nonzero terms, satisfies $P_{14}(A)>P_{15}(A)$. A sparse family shows that for every real $r>1$ some real symmetric positive definite matrix has a $q$-permanent with negative derivative at $q=r$. The counterexamples can be chosen to be correlation matrices without zero entries or, separately, to have integer entries. We also show that the smallest order of a counterexample is four, and that matrices supported on three coordinates separated by identity blocks fail much closer to $q=1$: in order five already near $q=1.7$, and in order $O(1/(r-1))$ at any prescribed $q=r>1$.
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