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,

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:

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:

For n=7:

+ 1/24 + 1/12 + 1/72 + 1/48 + 1/48 + 1/240 + 1/5040

And for n=11

+ 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