I’ve been reading a paper by Robert Morris (“Sets, Scales and Rhythmic Cycles; A Classification of Talas in Indian Music”) and came across a formula that I’ve found a bit tricky. He is referring to the “fragmentation” of a distribution and includes the formula below without derivation or reference. I’m pretty new to statistics, so this may be a standard formula that I’m just unaware of. However, I haven’t been able to find it in the same format online.

One feature for use in ordering talas is fragmentation. We have already grouped talas into partition classes. All talas in a particular partition class have the same fragmentation. We use the partition P as the input to a function that yields the fragmentation of the partition. Fragmentation varies between 0 and 1 and is a measure of the uniformity of a distributionâ€”the higher the fragmentation, the more even the distribution. We calculate the fragmentation of a partition of the number N into z parts using the following formula…:

$ FRAG(P)=1 – \frac{\sum_{k=1}^{z}{PAIRS(p_{k})}}{PAIRS(N)}$ where $ PAIRS(s)=\frac{{s^2}-s}{2} \:, \: P=\{{p_{1},p_{2}, p_{3}},…p_{z}\},\ N = sum(P), and \: z = card(p) . $

I found the formula to be much more readable in this format:

$ Let \: P = \{p_{1}, p_{2}, p_{3},…, p_{z}\}, \: z = card(P),\: and \: N = sum(P).\ FRAG(P)=1- \frac{\sum_{k=1}^{z} \frac{p_{k}^{2}-p_{k}}{2}}{\frac{N^{2}-N}{2}}=1-2\frac{\sum_{k=1}^{z}\frac{p_{k}^2-p_{k}}{2}}{N^2-N}$

The author uses the formula with the example $ P=\{2, 2, 4\} \rightarrow N = 2 + 2 + 4 = 8$ and $ z = 3.$ This returns $ FRAG(P)=1-2(\frac{8}{56})=0.714285714…$

Does this formula (or a similar one) have a name? Are there any places where I can find some further information? More generally, what does this mean?

Thanks for the help!