[EM] number of ways to vote a ranked ballot.

[Martin Harper]

Excellent. Thanks both of you for your help. :)

In case anyone is interested, I plugged that into Excel to calculate the 
first 25 results, and played around with formulas for a while: I settled 
on B(N)= int( N! x exp(N*B) x C ). With B=0.3666 and C=0.7213 (aprox). 
With the correct values of B and C I could fit to the first ten values 
perfectly. It began to overestimate from 11-25 by 10^7 or so, but that 
may well have been just inaccuracy in the spreadsheet's floating point 
calculations.

For 25 candidates, ranking with draws allows about 10^29 ways of marking 
a ballot paper, whereas without draws allows about 10^25 - so there's 
little difference between the two compared to the difference from 
Approval's 10^7 and Plurality's 25...

Thanks again.

Richard Moore wrote:

> I came up with a recursive formula and tested it against Rob's numbers. 
> It seems to work.
> For N candidates,
> 
> possible ballots = sum( f(N,m) )
> 
> The summation is over all integer values of m from 1 to N, inclusive. 
> The function
> f(N,m) represents the number of ways N objects can be put into m slots 
> without
> leaving any empty slots. The recursive definition of f(N,m) is:
> 
> f(N,1) = 1
> f(N,m) = N! for N=m
> f(N,m) = m( f(N-1,m) + f(N-1,m-1) ) for all other cases
> 
> Richard
> 
> 
> Rob LeGrand wrote:
> 
>> Martin wrote:
>> 
>>> Ok, if there are n candidats then there are n! ways to vote a fully 
>>> ranked ballot, and {int(e x n!)} ways to vote a truncated ballot, or 
>>> {int((e-1) x (n!) - 1)} ways if you count votes like A>B>C(>D) as 
>>> equivalent to A>B>C>D. All this I've found out by reading around 
>>> websites and such.
>>> 
>>> However, I can't seem to find anywhere which says how many ways there 
>>> are to vote a ranked ballot which allows draws in arbitrary places, and 
>>> I can't see any way to work it out. Any maths/stats people here know 
>>> what the answer is, or where I might find out?
>> 
>> 
>> I haven't been able to figure out a simple general formula, but here are my
>> results for up to 6 candidates:
>> 
>> candidates:          1    2    3    4    5    6
>> possible ballots:    1    3   13   75  541 4683
>> 
>> =====
>> Rob LeGrand
>> honky98 at aggies.org <mailto:honky98 at aggies.org>
>> http://www.aggies.org/honky98/
>> 
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