Asymptotics and statistics on Fishburn matrices: Dimension distribution and a conjecture of Stoimenow. Issue 2 (14th June 2022) - Hwang, Hsien‐Kuei; Jin, Emma Yu; Schlosser, Michael J.
Abstract: 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 {0, 1} or nonnegative integers ℕ 0 N_0 . The analytic saddle‐point approach we apply, based on a powerful transformation for q q ‐series due to Andrews and Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.