2020/12/25 by Hsien-Kuei Hwang, Hwang, Hsien-Kuei, Emma Yu Jin +3
Mathematics · #05A15 #05A16 #60C05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A16 #msc:60C05 #msc:60F05
paper · pdf · doi:10.48550/arxiv.2012.13570
35 pages, comments are welcome
arxiv created 2020/12/25 · arxiv updated 2020/12/29
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 Jelínek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.