The Smallest Parts Partition Function

Today I turned 25773 days old and I felt it shouldn't pass without making mention of this number's connection to the Smallest Parts Partition Function. This function assigns to each natural number \(n\), another number which is the total of the smallest parts in all partitions of \(n\). Here is the mapping, from \(n=1\) up to \(n=30\):

1, 3, 5, 10, 14, 26, 35, 57, 80, 119, 161, 238, 315, 440, 589, 801, 1048, 1407, 1820, 2399, 3087, 3998, 5092, 6545, 8263, 10486, 13165, 16562, 20630, 25773, ...

Figure 1 shows the SageMath code that I wrote to generate this sequence, up to and including 25773. Here is the Permalink.

Figure 1: SageMath code to generate
Smallest Parts Partition Numbers

Here is the example given in the OEIS A092269 comments:

Partitions of 4 are [1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]. 

1 appears four times in [1, 1, 1, 1]

1 appears two times in [1, 1, 2]

2 appears two times in [2, 2]

1 appears once in [1, 3]

4 appears once in [4]

Thus a(4)=4+2+2+1+1=10

Figure 2 shows a plot of the values up to 25773:

Figure 2: plot of the Smallest Parts Partition Function

Like the partition function, there is a generating function but it's rather complicated and I won't include it here. However, it can be viewed in the OEIS comments. I just wanted to mention it because the next member of the sequence is 31897 which is a long way off. There are a number of academic papers about this function so it is a topic of serious mathematical interest.

The number of partitions from 1 to 30 are shown in the list below:

[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]

It's interesting to look at the ratio between the total value of the number of smallest parts and the number of partitions for numbers between 1 and 30. The results and a plot of these values can be found in Figure 3.

Figure 3

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