Bell numbers in Matsunaga's and Arima's Genjik\=o combinatorics: Modern perspectives and local limit theorems

Abstract

We examine and clarify in detail the contributions of Yoshisuke Matsunaga (1694?--1744) to the computation of Bell numbers in the eighteenth century (in the Edo period), providing modern perspectives to some unknown materials that are by far the earliest in the history of Bell numbers. Later clarification and developments by Yoriyuki Arima (1714--1783), and several new results such as the asymptotic distributions (notably the corresponding local limit theorems) of a few closely related sequences are also given.