# A theorem about partitioning consecutive numbers

## TL;DR

This paper explores a new property of triangular numbers related to their partitions into consecutive numbers, linking these partitions to partitions of the sequence starting at 1, building on Sylvester's classical result.

## Contribution

It introduces a novel statement connecting partitions of triangular numbers into consecutive integers with partitions of the initial sequence, extending classical partition theory.

## Key findings

- Establishes a correspondence between partitions of triangular numbers and partitions of initial sequences.
- Provides a new perspective on partitioning numbers into consecutive integers.
- Builds on Sylvester's theorem to develop a broader combinatorial insight.

## Abstract

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an interesting statement about triangular numbers, those positive integers which can be partitioned into consecutive numbers beginning at 1. For every partition of a triangular number n into consecutive numbers we can partition the sequence of numbers beginning at 1, adding up to n again, such that every part of this partition adds up to exactly one number of the chosen partition of n.