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

TL;DR

This paper explores the historical development of Bell numbers through Matsunaga's and Arima's work, providing modern insights and new asymptotic results, including local limit theorems, on related combinatorial sequences.

Contribution

It offers a detailed historical analysis of early Bell number computations and presents new asymptotic and local limit theorems for related sequences.

Findings

Historical clarification of Matsunaga's contributions

New asymptotic distribution results for related sequences

Local limit theorems established for combinatorial sequences

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.