This may be more about math syntax than math. I am uncertain about how to write pi and sum notations when you are not "looping" through a set of consecutive numbers, but rather through a set of implicit cases. So rather than this: $$\prod^n_{i=0}(something),\qquad i=0,1,2,3, ..., n$$
I would like $i$ to refer to a table for example (not sure of correct notation):
$$\prod_{i}(something),\qquad i=0,12,4,5 \;\;(explicitly\;stated\;values)$$
How could I express this in the pi notation? Is the above notation correct?
An example:
In my example we have a molecule with a number of atoms $n_e$ of a certain element $e$, and a number $n_{i_e}$ of each isotope $i_e$ of that element:
$$n_e!\prod_{i_e}\frac{1}{n_{i_e}!}$$
And for all elements $e$:
$$\prod_{e}n_e!\prod_{i_e}\frac{1}{n_{i_e}!}$$
This formula is used for calculating the numbers of configurations a molecule with this composition has. So imagine a completely made up molecule C16O4 with these isotopes:
Only taking carbon into consideration the number would be:
$$n_e!\prod_{i_e}\frac{1}{n_{i_e}!}=(14+2)!\frac{1}{14!2!}$$
And if we take all elements into consideration:
$$\prod_{e}n_e!\prod_{i_e}\frac{1}{n_{i_e}!}=(14+2)!\frac{1}{14!2!}(3+1+0)!\frac{1}{3!1!0!}$$
I would like $\prod_e$ to say "For all elements" and $\prod_{i_e}$ to say "For all isotopes in element e". Is this syntax right?
I am no professor in mathematics but I have seen (in many different places) that one writes text under the summation symbol exactly as you wrote above. Something like $$ \sum_{all \ x \ such \ that ...}=... $$ as you can see this easily creates "ugly" expressions but if you keep your text short it is definitely viable. If you would like to use a certain sequence of numbers as indexes you can give the definition of the sequence like $$ \sum_{x=2k+3,\ k\in\mathbb{Z}}=... $$ I hope this helped :)