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Today, I found an interesting identity. First, let the number of ways of partitioning a set of n elements into m nonempty sets be S(n,m). Also, let

Eqn 1

This can be seen either as the Taylor Series of f(x) or the generating function of B_n

The identity would be:

Eqn 2.png

A special case of this identity would be:

eqn 3.png

Where e is euler’s constant or e=2.71828182846…

It looks really complicated with all the notation but in essence, the identity can be easily understood. I will explain the summation notation in a future post regarding Analytic Number Theory.

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