Asymptotics and statistics on Fishburn matrices: Dimension distribution and a conjecture of Stoimenow. Issue 2 (14th June 2022)

• Record Type: Journal Article
• Authors: Hwang, Hsien‐Kuei; Jin, Emma Yu; Schlosser, Michael J.
• Is Part Of: Random structures & algorithms. Volume 62:Issue 2(2023)
• Page Start: 479
• Page End: 518
• Publication Date: 2022-06-14
• Subjects: a conjecture of Stoimenow -- dimension -- Fishburn matrices -- q$$ q $$‐series -- saddle‐point method; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.21100
• Languages: English
• ISSNs: 1042-9832

Abstract

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 } $$ \left\{0, 1\right\} $$ or nonnegative integers ℕ 0 $$ {\mathbb{N}}_0 $$ . 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.