Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester (Q1644634)

This paper reviews results relating partitions of integers to trapezoidal numbers and the divisor function. A \(k\)-trapezoid with difference t is the sum of some finite arithmetic progression \(\{a,a+t,\dots,a+(k-1)t\}\) with \(k \in \mathbb{N}\), \(t \in \mathbb{N}_0\), \(a \in \mathbb{Z}\). A \(k\)-trapezoid is a number which is a trapezoid with \(k\) parts and difference \(t=1\). For example, the 2-trapezoids are precisely the odd numbers. Let \(d_1(n)\) be the number of positive odd divisors of \(n\), and \(d(n,\theta)\) be the number of positive divisors of \(n\) less than \(\theta\). A sampling of theorems covered in the paper includes the following. Theorem. Let \(t\) be an odd positive integer. For every odd positive divisor \(k\) of \(n\), there is exactly one representation of \(n\) as a \(k\)-trapezoid with difference \(t\), and exactly one representation of \(n\) as a \((2n/k)\)-trapezoid with difference \(t\). The number of representations of \(n\) as a trapezoid with difference \(t\) is \(2d_1(n)\). Theorem. [\textit{G. E. Andrews} et al., Duke Math. J. 108, No. 3, 395--419 (2001; Zbl 1005.11048)] The number of representations of \(n\) as the sum of an even number of consecutive positive integers, minus the number of representations of \(n\) as the sum of an odd number of consecutive positive integers, is \(d(n,\sqrt{2n}) - d(n,\sqrt{(n/2)})\). In Sections 4 through 6, the author gives a lucid proof of Sylvester's algorithm for a bijection between partitions into odd parts and partitions into distinct parts, and proves that this maps partitions with exactly \(\ell\) odd parts to partitions into distinct parts with \(\ell\) maximal subsequences of consecutive integers, which can be considered \(k\)-trapezoids with difference 1. The paper closes posing a problem: observe that Euler's odd-distinct theorem matches a partition into 2-trapezoids (odd numbers) with a partition into distinct parts. What partitions correspond to partitions into \(k\)-trapezoids for \(k \geq 3\)? For the entire collection see [Zbl 1388.11003].