Various analytic observations on combinations

TL;DR

This paper explores the mathematical properties of partitions, symmetric functions, and infinite series, introducing new relations and recurrences, and discussing the pentagonal number theorem, contributing to the foundational understanding of partition theory.

Contribution

Euler introduces new relations among symmetric functions, infinite series, and partitions, and states the pentagonal number theorem with a partial proof, advancing combinatorial mathematics.

Findings

Derived relations between series and products of symmetric functions

Proved recurrence relations for partitions with and without repetition

Stated the pentagonal number theorem, noting incomplete proof

Abstract

E158 in the Enestrom index. Translation of the Latin original "Observationes analyticae variae de combinationibus" (1741). This paper introduces the problem of partitions, or partitio numerorum (the partition of integers). In the first part of the paper Euler looks at infinite symmetric functions. He defines three types of series: the first denoted with capital Latin letters are sums of powers, e.g. $A=a+b+c+...$, $B=a^2+b^2+c^3+...$, etc.; the second denoted with lower case Greek letters are the elementary symmetric functions; the third denoted with Germanic letters are sums of all combinations of $n$ symbols, e.g. $\mathfrak{A}=a+b+c+...$ is the series for $n=1$, $\mathfrak{B}=a^2+ab+b^2+ac+bc+c^2+...$ is the series for $n=2$, etc. Euler proves a lot of relations between these series. He defines some infinite products and proves some more relations between the products and these series. Then in \S 17 he looks at the particular case where $a=n,b=n^2,c=n^3$ etc. In \S 19 he says the Naud\'e has proposed studying the number of ways to break an integer into a certain number of parts. Euler proves his recurrence relations for the number of partitions into a $\mu$ parts with repetition and without repetition. Finally at the end of the paper Euler states the pentagonal number theorem, but says he hasn't been able to prove it rigorously.