[EM] IRV ballot pile count (proof of closed form)

[Kristofer Munsterhjelm]

Kathy Dopp wrote:
> People on this list seem to still be sending around their incorrect or
> incomplete formulas for the number of possible rank orders for rank
> order ballots.  This number BTW does *not* correspond to the number of
> piles needed to count IRV which is a lesser number but does correspond
> to the only method of making IRV precinct-summable.

For precinct summability, whether or not you include both "A>B" and 
"A>B>C" votes as distinct (in a three-candidate election) doesn't really 
matter because the factorial term dominates and so one can broadly say:

- When one formula says it's practical to send raw ballot counts around, 
it's practical to do it by any of the other formulas

- When not practical for one, it's not practical for the others either.

Of course, when verifying the outcomes, you'd want to have the exact 
number right, but it seems to me that the question of whether the method 
can be feasibly summed in precincts by transmitting raw counts ("how 
many voted this order, how many voted that order") does not depend on 
the exact nature of the formula, because they all grow so quickly.