Let's say you've got a bunch of parentheses - , to be exact. And these are really awesome parentheses, too. They're so awesome, you simply can't separate a pair. So for each (, you must have a corresponding ). A valid sequence would be something like (()()(())), but ))()( isn't.

Here's the problem: How many groupings do we have for the values of ?

Well, then. What first? How about making this problem all nice and math-y?

We can say that a group of parentheses is valid if the number of ( equals the number of ) [darn that looks funny, and I will henceforth be making all parenthetical comments in brackets for clarity] and that as we read from left to right, we add 1 for each ( and subtract 1 from each ), and the final number should be nonnegative.

There. All math-ed up.

Hmm. We still don't have much a lead, so let's try listing out all the possibilities for, say, through .

: Obviously, we have exactly 1 way to do this.
: () = 1 way
: ()(), (()) = 2 ways
[from here on out, I won't list the combinations, just the number of ways, because this is highly painful to type]
: 5 ways
: 14 ways

Let's see: 1, 1, 2, 5, 14. You might recognize these numbers! But for those of you that don't...

Tune in next week for part 2!

www.mathsnet.net

There are actually some pretty interesting applets in there. Have fun.

This is the first non-mathematical entry of this blog, commemorating Mr. Holbrook, the greatly esteemed coach of the AAST math team. You devoted so much of your time and money for us, and we were all shocked when you died, to the point that I thought Fanfan was making a cruel joke...

Rest in peace.