Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow

TL;DR

This paper proves the asymptotic normality of the dimension distribution in large Fishburn matrices and addresses a related size distribution problem, using complex analysis and a transformation for q-series, also solving a conjecture in Vassiliev invariants.

Contribution

It introduces a general framework for Fishburn matrices, establishing their asymptotic distributions and solving two open combinatorial questions with advanced analytic methods.

Findings

Dimension of large Fishburn matrices is asymptotically normal.

Size distribution under large dimension follows a quadratic normal limit law.

The methods solve a conjecture of Stoimenow in Vassiliev invariants.

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 binary or nonnegative integers. The analytic saddle-point approach we apply, based on a powerful transformation for $q$-series due to Andrews and Jel\'inek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.