Partitions

As I've been reading "The Man Who Knew Infinity", the topic of partitions came up and I felt impelled to delve a little further into the topic. Wikipedia defines a partition as follows:

In number theory and combinatorics, a partition of a positive integer \(n\), also called an integer partition, is a way of writing \(n\) as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, 4 can be partitioned in five distinct ways: 

4 

3 + 1 

2 + 2 

2 + 1 + 1 

1 + 1 + 1 + 1 

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1. 

A summand in a partition is also called a part. The number of partitions of \(n\) is given by the partition function \(p(n) \). So \(p(4)\) = 5. The notation \( \lambda \vdash n \) means that \( \lambda \) is a partition of \( n \). 

Partitions can be graphically visualised with Young diagrams or Ferrers diagrams.

Wolfram Alpha can be used to generate the number of partitions for a given number. An example of the number 5 is shown below, with visualisation using Ferrers diagrams:

 

The OEIS sequence for the number of partitions of the various numbers is A000041 and begins as follows:

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525

In terms of my numbered days, the last partition number was 21637 and the next will be 26015. This is only the briefest of introductions to the topic and I've not even mentioned the partition-generating function. I'll add it here just for completeness:$$\sum_{n=0}^{\infty} p(n) \, x^n=\prod_{k=1}^{\infty} \left ( \frac{1}{1-x^k} \right )$$

on Thursday April 8th 2021