Posted by Jason Polak on 07. April 2018 · Write a comment · Categories: number-theory · Tags: ,

There is a cool way to express 1 as a sum of unit fractions using partitions of a fixed positive integer. What do we mean by partition? If $n$ is such an integer then a partition is just a sum $e_1d_1 + \cdots + e_kd_k = n$ where $d_i$ are positive integers. For example,

7 = 4 + 1 + 1 = 4 + 2(1)

The order of the partition is not very interesting and so we identify partitions up to order. Thus, here are all the 15 partitions of 7:

7
6+1
5+2
5+1+1
4+3
4+2+1
4+1+1+1
3+3+1
3+2+2
3+2+1+1
3+1+1+1+1
2+2+2+1
2+2+1+1+1
2+1+1+1+1+1
1+1+1+1+1+1+1

Group the same numbers together so that each partition is written as $n = \sum e_id_i$ where there are $e_i$ appearances of $d_i$ (or vice-versa, it’s symmetric). Then it’s a theorem that:
$$
1 = \sum (e_1!\cdots e_k!\cdot d_1^{e_1}\cdots d_k^{e_k})^{-1}.$$
This partition identity has a bunch of proofs. A neat one appears in the paper “Using Factorizations to Prove a Partition Identity” by David Dobbs and Timothy Kilbourn. In their proof, they used an asympotitc expression for the number of irreducible polynomials over a finite field of a given degree $n$ (the same $n$ that appears in the partition).

Here are some examples of this identity. For n=5, we have:

1 = 1/5 + 1/4 + 1/6 + 1/6 + 1/8 + 1/12 + 1/120

For n=7:

1 = 1/7 + 1/6 + 1/10 + 1/10 + 1/12 + 1/8 + 1/24 + 1/18
+ 1/24 + 1/12 + 1/72 + 1/48 + 1/48 + 1/240 + 1/5040

And for n=11

1 = 1/11 + 1/10 + 1/18 + 1/18 + 1/24 + 1/16 + 1/48 + 1/28 + 1/21
+ 1/56 + 1/28 + 1/168 + 1/30 + 1/24 + 1/36
+ 1/36 + 1/48 + 1/72 + 1/720 + 1/50 + 1/40 + 1/40
+ 1/90 + 1/30 + 1/90 + 1/240 + 1/80 + 1/240
+ 1/3600 + 1/96 + 1/64 + 1/192 + 1/72 + 1/96 + 1/48
+ 1/288 + 1/192 + 1/192 + 1/960 + 1/20160 + 1/324
+ 1/324 + 1/144 + 1/216 + 1/2160 + 1/1152 + 1/288 + 1/576
+ 1/4320 + 1/120960 + 1/3840 + 1/2304
+ 1/5760 + 1/40320 + 1/725760 + 1/39916800

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