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

[Kristofer Munsterhjelm]

robert bristow-johnson wrote:
> 
> On Feb 2, 2010, at 2:28 PM, robert bristow-johnson wrote:
> 
>>
>> Warren tells me that
>>
>>     C-1
>>     SUM{ C!/n! }
>>     n=1
>>
>> has a closed form, but didn't tell me what it is.  does someone have 
>> the closed form for it?  i fiddled with it a little, and i can 
>> certainly see an asymptotic limit of
>>
>>     (e-1)(C!)
>>
>> as C gets large, but i don't see an exact closed form for it.  if 
>> someone has such a closed form, would you mind sharing it?
> 
> Okay, I spent a little time working on this and figgered it out.  The 
> fact that the number of distinct piles needed to represent all possible 
> manners of *relatively* ranking C candidates (no ties except unranked 
> candidates are tied for lowest rank) is
> 
>     C-1
>     SUM{ C!/n! }  =  floor( (e-1) C! ) - 1
>     n=1

Now I wonder if there's a closed form for the number of orders with both 
equality and truncation permitted. Since I don't quite get the proof, I 
can't answer, though!