On the partition of numbers into parts of a given type and number

TL;DR

Euler's 1768 paper explores the number of partitions of integers into parts of specific types, mainly focusing on recurrence formulas for partitions involving dice, aiming to clarify and extend his earlier work on partitions.

Contribution

The paper compiles recurrence formulas for partitions into specific parts, emphasizing special cases and clarifying existing theories without introducing new results.

Findings

Recurrence formulas for partitions into specific parts

Analysis of partitions involving dice throws

Clarification of earlier partition theories

Abstract

E394 in the Enestrom index. Translated from the Latin original, "De partitione numerorum in partes tam numero quam specie datas" (1768). Euler finds a lot of recurrence formulas for the number of partitions of $N$ into $n$ parts from some set like 1 to 6 (numbers on the sides of a die). He starts the paper talking about how many ways a number $N$ can be formed by throwing $n$ dice. There do not seem to be any new results or ideas here that weren't in "Observationes analyticae variae de combinationibus", E158 and "De partitione numerorum", E191. In this paper Euler just does a lot of special cases. My impression is that Euler is trying to make his theory of partitions more approachable,. Also, maybe for his own benefit he wants to say it all again in different words, to make it clear.