Using Factorizations to Prove a Partition Identity

This equation may not seem remarkable at first glance. After all, any positive rational number can be expressed as a sum of distinct unit fractions [11, p. 261], and 1 can be so expressed more easily, as in the equation 16 +

1 2 = 1. However,

the first equation is special, for its summands can be written as 1

5! · 15 , 1

3! · 1! · 13 · 21 , 1

2! · 1! · 12 · 31 , 1

1! · 2! · 11 · 22 , 1

1! · 1! · 11 · 41 , 1

1! · 1! · 21 · 31 , 1

1! · 51 .