# [EM] IRV vs Plurality (back to the pile count controversy)

James Gilmour jgilmour at globalnet.co.uk
Thu Jan 21 15:30:56 PST 2010

```robert bristow-johnson  > Sent: Thursday, January 21, 2010 6:49 AM
> but breaking it down to piles regarding every conceivable permutation
> of candidate preference is *still* breaking it down to a finite
> number of piles.  for 3 candidates, that number is 9.  if you or
> Kathy say it's 15, then you're wrong (and it's your slip that's
> showing).  for 4 candidates the number of necessary piles is
> 40.
>  for  N candidates, the number of piles necessary, P(N) is
>
>             N-1
>      P(N) = SUM{ N!/n! }
>             n=1
>
> not
>
>             N-1
>      P(N) = SUM{ N!/n! }
>             n=0

I do not intend to comment on your formula, but I calculate the numbers of possible unique preference profiles for increasing
numbers of candidates (N) as follows:

N	Unique Preference Profiles
2	4
3	15
4	64
5	325
6	1,956
7	13,699
8	109,600
9	986,409
10	9,864,100
11	108,505,111
12	1,302,061,344
13	16,926,797,485
14	236,975,164,804
15	3,554,627,472,075
16	56,874,039,553,216
17	966,858,672,404,689
18	17,403,456,103,284,400
19	330,665,665,962,404,000
20	6,613,313,319,248,080,000

Where there are large numbers of candidates, the maximum possible number of unique preference profiles will be limited by the number
of voters.  Thus if there are 10,000 valid votes and 12 candidates, the maximum possible number of preference profiles would be
10,000 and not 1,302,061,344.

In practice the actual number of preference profiles would be even lower, as significant numbers of voters would record identical
patterns of preferences.  Thus in the Meath constituency for the Dáil Éireann election in 2002 with 14 candidates (236,975,164,804
possibilities), there were 64,081 valid votes, but only 25,101 unique preference profiles.

The Minneapolis STV (RCV) ballots were all hand sorted to unique preference profiles for each precinct and hand counted.  This was
unnecessary but feasible as the voters could not record more than three preferences (rankings), no matter the numbers of candidates.
I understand the full preference profiles, probably at precinct level, will be published on the City website, but they are not there
yet.

James Gilmour

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