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Mon, 05 Mar 2007
An integer partition puzzle
Here's one interesting fact: it's quite easy to calculate the number of partitions of N. Let P(n, k) be the number of partitions of n into parts that are at least k. Then it's easy to see that:
$$P(n, k) = \sum_{i=k}^{n-1} P(n-i, k)$$ And there are simple boundary conditions: P(n, n) = 1; P(n, k) = 0 when k > n, and so forth. And P(n), the number of partitions of n into parts of any size, is just P(n, 1). So a program to calculate P(n) is very simple:
my @P; sub P { my ($n, $k) = @_; return 0 if $n < 0; return 1 if $n == 0; return 0 if $k > $n; my $r = $P[$n] ||= []; return $r->[$k] if defined $r->[$k]; return $r->[$k] = P($n-$k, $k) + P($n, $k+1); } sub part { P($_[0], 1); } for (1..100) { printf "%3d %10d\n", $_, part($_); }I had a funny conversation once with someone who ought to have known better: I remarked that it was easy to calculate P(n), and disagreed with me, asking why Rademacher's closed-form expression for P(n) had been such a breakthrough. But the two properties are independent; the same is true for lots of stuff. Just because you can calculate something doesn't mean you understand it. Calculating ζ(2) is quick and easy, but it was a major breakthrough when Euler discovered that it was equal to π^{2}/6. Calculating ζ(3) is even quicker and easier, but nobody has any idea what the value represents. Similarly, P(n) is easy to calculate, but harder to understand. Ramanujan observed, and proved, that P(5k+4) is always a multiple of 5, which had somehow escaped everyone's notice until then. And there are a couple of other similar identities which were proved later: P(7k+5) is always a multiple of 7; P(11k+6) is always a multiple of 11. Based on that information, any idiot could conjecture that P(13k+7) would always be a multiple of 13; this conjecture is wrong. (P(7) = 15.) Anyway, all that is just leading up to the real point of this note, which is that I was tabulating the number of partitions of n into exactly k parts, which is also quite easy. Let's call this Q(n, k). And I discovered that Q(13, 4) = Q(13, 5). There are 18 ways to divide a pile of 13 beans into 4 piles, and also 18 ways to divide the beans into 5 piles.
So far, I haven't turned anything up; it seems to be a coincidence. A simpler problem of the same type is that Q(8, 3) = Q(8, 4); that seems to be a coincidence too:
Oh well, sometimes these things don't work out the way you'd like.
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