Journey to Integer Partitions

Introduction

After 114 meetings and 75 hours of studying together, our analytic number theory book discussion group has finally reached the final chapter of the book Introduction to Analytic Number Theory (Apostol, 1976). We have less than 18 pages to read in order to complete reading this book. Considering that we meet 3-4 times in a week and we discuss about 2-3 pages in every meeting, it appears that we would be able to complete reading this book in another 2 weeks.

Reading this book has been quite a journey! The previous three posts on this blog provide an account of how this journey has been. It has been fun, of course. The best part of hosting a book discussion group like this has been the number of extremely smart people I got an opportunity to meet and interact with. The insights and comments on the study material that others shared during the meetings were very helpful.

The meeting log shows that our meetings started really small with only 4 participants in the first meeting in March 2021 and then it gradually grew to about 10-12 regular members within a month. Then a few months later, the number of participants began dwindling a little. This happened because some members of the group had to drop out as they got busy with other personal or professional engagements. However, six months later, we still have about 4-5 regular participants meeting consistently. I think it is pretty good that we have made it this far.

Unrestricted Partitions

The final chapter on integer partitions is very unlike all the previous 12 chapters. While the previous chapters dealt with multiplicative number theory, this final chapter deals with additive number theory. For example, the first theorem talks about an interesting property of unrestricted partitions. We study the number of ways a positive integer can be expressed as a sum of positive integers. The number of summands is unrestricted, repetition of summands is allowed, and the order of the summands is not taken into account. For example, the number 3 has 3 partitions: 3, 2 + 1, and 1 + 1 + 1. Similarly, the number 4 has 5 partitions: 4, 3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 + 1 + 1.

I have always wanted to learn about partitions more deeply, so I am quite happy that this book ends with a chapter on partitions. The subject of partitions is rich with very interesting results obtained by various accomplished mathematicians. In the book, the first theorem about partitions is a very simple one that follows from the geometric representation of partitions. Let us see an illustration first.

How many partitions of 6 are there? There are 11 partitions of 6. They are 6, 5 + 1, 4 + 2, 4 + 1 + 1, 3 + 3, 3 + 2 + 1, 3 + 1 + 1 + 1, 2 + 2 + 2, 2 + 2 + 1 + 1, 2 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1. Now how many of these partitions are made up of 5 parts? Each summand is called a part. The answer is 2. There are 2 partitions of 6 that are made up of 5 parts. They are 3 + 1 + 1 + 1 and 2 + 2 + 1 + 1. Let us represent both these partitions as arrangements of lattice points. Here is the representation of the partition 3 + 1 + 1 + 1:

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Now if we read this arrangement from left-to-right, column-by-column, we get another partition of 6, i.e., 4 + 1 + 1. Note that the number of parts in 3 + 1 + 1 + 1 (i.e., 4) appears as the largest part in 4 + 1 + 1. Similarly, the number of parts in 4 + 1 + 1 (i.e., 3) appears as the largest part in 3 + 1 + 1 + 1. Let us see one more example of this relationship. Here is the geometric representation of 2 + 2 + 1 + 1:

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Once again, reading this representation from left-to-right, we get 4 + 2, another partition of 6. Once again, we can see that the number of partitions in 2 + 2 + 1 + 1 (i.e., 4) appears as the largest part in 4 + 2, and vice versa. These observations lead to the first theorem in the chapter on partitions:

Theorem 14.1 The number ofpartitions of \( n \) into \( m \) parts is equal to the number of partitions of \( n \) into parts, the largest of which is \( m. \)

That was a brief introduction to the chapter on partitions. In the next two or so weeks, we will dive deeper into the theory of partitions.

Next Meeting

If this blog post was fun for you, consider joining our next meeting. Our next meeting is on Tue, 21 Sep 2021 at 17:00 UTC. Since we are at the beginning of a new chapter, it is a good time for new participants to join us. It is also a good time for members who have been away for a while to join us back. Since this chapter does not depend much on the previous chapters, new participants should be able to join our reading sessions for this chapter and follow along easily without too much effort.

To join our discussions, see our channel details in the main page here. To get the meeting link for the next meeting, visit the analytic number theory book page.

It is worth mentioning here that lurking is absolutely fine in our meetings. In fact, most participants of our meetings join in and stay silent throughout the meeting. Only a few members talk via audio/video or chat. This is considered absolutely normal in our meetings, so please do not hesitate to join our meetings!