Asymptotics and statistics on Fishburn matrices: Dimension distribution and a conjecture of Stoimenow

We establish the asymptotic normality of the dimension of large‐size random Fishburn matrices by a complex‐analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to{ 0 , 1 } $$ \left\{0,1\right\} $$ or nonnegative integersℕ 0$$ {\mathbb{N}}_0 $$ . The analytic saddle‐point approach we apply, based on a powerful transformation forq $$ q $$ ‐series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.