The mathematical studies of G.W. Leibniz on combinatorics

Abstract

Leibniz considered the “ars combinatoria” as a science of fundamental significance, much more extensive than the combinatorics of today. His only publications in the field were his youthful Dissertatio de Arte Combinatoria of 1666 and a short article on probability, but he left an extensive (hitherto unpublished and unstudied) Nachlass dealing with five related topics: the basic operations of combinatorics, symmetric functions in connection with theory of equations, partitions (additive theory of numbers), determinants, and theory of probability and related fields. This paper concentrates on the first and third topics as they appear in published sources and the Nachlass. It shows that Leibniz was in possession of many results not published by other mathematicians until many decades later. These include a recursion formula for partitions of n into k parts (first published by Euler in 1751), the Stirling numbers of the second kind (first published in 1730), and several special cases of the general formula for partitions that was published only in 1840 by Stern.