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Asymptotics and statistics on Fishburn matrices: Dimension distribution and a conjecture of Stoimenow
Author(s) -
Hwang HsienKuei,
Jin Emma Yu,
Schlosser Michael J.
Publication year - 2023
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21100
Subject(s) - mathematics , conjecture , dimension (graph theory) , transformation (genetics) , saddle , asymptotic distribution , combinatorics , statistics , biochemistry , chemistry , estimator , gene , mathematical optimization
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.