Closed-form enumeration of totally positive matrices

Derive a simple closed-form expression for the number of totally positive matrices in M_{m,n}(F_q) when F_q has odd characteristic and max{m,n} is at least 3, or when F_q has characteristic 2 and both m and n are at least 3.

Background

The paper obtains exact enumeration formulas only in several low-dimensional cases and derives structural formulas involving algebraic-number powers for general dimensions. These structural formulas follow from representing the relevant conditions by algebraic sets and applying rationality of Weil zeta functions.

The authors explicitly state that a simple closed form remains unavailable in the indicated higher-dimensional regimes.

References

At present, we do not have a simple closed-form expression for the number of totally positive matrices in $M_{m,n}(\mathbb{F}_q)$ when $\mathbb{F}_q$ is of odd characteristic and $\max{m,n} \geq 3$, or when $\mathbb{F}_q$ is of characteristic $2$ and $m,n \geq 3$.

— Positive definite, positive semidefinite and totally positive matrices over finite fields  (2608.17702 - Ayyer et al., 18 Aug 2026) in Section “Structural formula” of Section “Totally positive matrices”