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arXiv · 1601.07058

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

Melvyn B. Nathanson
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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.

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Melvyn B. Nathanson. 2016-01-26. Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester. https://arxiv.org/abs/1601.07058

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