Simpson's diversity index

Suppose $R(\pi_k) = {1 - \pi_k}$, the probability that the next species you encounter is different from the one you have just seen. Then

$$\begin{eqnarray*} \Delta_1(C) &=& \sum_{k=1}^s\pi_k(1-\pi_k) \cr &=& 1 - \sum_{k=1}^s\pi_k^2 \end{eqnarray*}$$

For those of you who've had population genetics, you can think of the $\pi_k$ as allele frequencies and Simpson's diversity as the panmictic heterozygosity. It's the probability that any two individuals chosen at random from the community belong to different species.