Misc,
Various analytic observations on combinations
(November 2007)
Abstract
E158 in the Enestrom index. Translation of the Latin original Öbservationes
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. \$A=a+b+c+...\$ is the
series for \$n=1\$, \$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 Naudé 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.
