The All-Thing

All stick and no carrot, since ought-three.


三衢道中 (曾幾)


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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:

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).

Posted at 13:38 | /math | (leave a comment) | permalink

Gambler's Ruin Solution

My language exchange partner and I were going over the Gambler's Ruin problem yesterday (he is a big random walk guy and yes, this is the type of stuff we end up talking about in the English portion) and we came up with a solution remarkably similar to this one I found today:

No random walks involved, just straight recurrence solving. Interesting that there's a hidden q != p assumption in there (which we didn't notice and had us scratching our heads as to why everything collapsed to 0/0 in the q = p case.)

Posted at 13:30 | /math | (leave a comment) | permalink


I swear to god I just heard a Grappelli sample in the middle of some random hiphop track.

Posted at 13:23 | /media/music | (leave a comment) | permalink


My only love sprung from my only hate! Too early seen unknown, and known too late! -- William Shakespeare, "Romeo and Juliet"