The All-Thing

All stick and no carrot, since ought-three.

各位能夠讀中文得來賓您好。小的這還在學中文中,恐怕中文寫得不太好,希望你們還看得懂。


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Thu, 09 Oct 2003
Whitney Numbers

This was cool. I've been working on a problem at work and at one point we needed to find the maximum number of ways of dividing an /n/-dimensional space into k partitions. It's easy enough to figure this out for the one- and two-dimensional case, and pretty mind-bending for the three-dimensional case, but what about the generalization?

So we go to the Online Encyclopedia of Integer Sequences, type in (get this) 2, 4, 8 (the first three entries for the 3-d case) and lo and behold, we get:

http://www.research.att.com/projects/OEIS?Anum=A004070

Cool or what? (A complete fluke as their sequence is the table read by anti-diagonals... wtf?)

So anyways, the solution is

W(n,k)=if k=0 or n=0 then 1 else W(n,k-1)+W(n-1,k-1), or
W(n,k)=Sum(binomial(k,i), i=0..n)

if you were curious (so order exponential, unfortunately for us).

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