[Foundation-l] Raw data of 2009 Board election ballots

[Aryeh Gregor]

On Wed, Aug 26, 2009 at 11:09 AM, Gregory Maxwell<gmaxwell at gmail.com> wrote:
> Nitpicking, but the number of possible unique ballots is much greater
> than the factorial because of equality, and equality must be preserved
> in order produce the election calculations. The formula mostly easily
> represented is a messy multipart recursive formula, which I'll spare
> you (in part because I don't know that I have all the boundary
> conditions right).  It's less than X!*2^(X-1).

I think it's the sum from k = 1 to n of S(n, k)*k!, where S(n, k) is a
Stirling number of the second kind.

http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind

S(n, k) is the number of ways to divide an n-element set into k
partitions.  So there are S(n, k) ways to obtain k distinct "levels"
of candidates after tying, and then there are k! different ways to
order the levels against each other.  Maxima gives the following
results:

(%i3) sum(stirling2(18,k)*(k!), k, 1, 18);
(%o3)                         3385534663256845323
(%i4) 18!;
(%o4)                          6402373705728000

So it's about 529 times more than 18!.  Admittedly, determining this
was a completely pointless exercise, but it was kind of fun.