Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester

TL;DR

This paper explores the relationship between trapezoidal numbers, divisor functions, and a partition theorem of Sylvester, providing a comprehensive review and a complete proof of the theorem's stratification of partitions.

Contribution

It offers a detailed analysis of how trapezoidal numbers relate to partitions and divisors, including a full proof of Sylvester's partition stratification theorem.

Findings

Established the connection between trapezoidal numbers and divisor functions.

Provided a complete proof of Sylvester's partition stratification theorem.

Clarified the relationship between partitions into odd parts and trapezoidal partitions.

Abstract

A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of consecutive integers, or, more generally, whose parts form a finite arithmetic progression. This paper reviews the relation between trapezoidal numbers, partitions, and the set of divisors of a positive integer. There is also a complete proof of a theorem of Sylvester that produces a stratification of the partitions of an integer into odd parts and partitions into disjoint trapezoids.