The Life of a Mathematician

So, it's been a while since I've posted. This is because I was catching up in my classes, and then taking finals. That's about all I did for the past however long, so there really wasn't much to post anyway. I'm going to be doing qualifying exams in August probably, so I have to get ready for them. Written in Algebra and oral over everything.

So, let's see...highlights. Starting on about Tuesday of last week, finals week, I had a sore throat which gradually got worse. This was nice timing for the two hour presentation I had to give in Representation Theory in Friday. Fortunately I was still able to talk at that time.

Friday afternoon I was planning to head back home for my cousin's high school graduation, but I missed the bus. I thought it left at 16:15, but it actually left at 16:10. So I got there a couple minutes early and was actually about three minutes late. I wasn't feeling great, so it was probably just as well.

Friday night I ended up watching the last two episodes of Enterprise. They were good, but it was disappointing that they couldn't play the series out for a few more seasons.

Saturday night, we had a party for all the local college graduates from the church. Afterwards we went to Greg's house and watched Episode II, but I slept through most of it. Then I went home and slept for about 12 hours straight. I didn't make it to church and I didn't play Ultimate Frisbee, but I feel a little bit better. I still can't talk well.

Next Thursday a bunch of us are going to go watch Episode III. Should be fun. Not sure when I'm going to see Hitchhiker's.

Meanwhile, I need a summer job.

Okay, enough about life; let's talk about math. Part of the stuff my presentation was on had a connection with combinatorics. Without getting into what these shapes are for in Representation Theory, consider shapes formed by connected squares with the following requirements:

-) The rows are all left-justified.
-) Each row is as long or longer than the one below it.

Examples (using *'s instead of squares):

***
**
**

*
*

****
***
***
*

Now, you can gradually build all the possibilities by starting with a single * and considering all the possible ways you can add one more at each level. Thus:

1:

*

2:

**

*
*

3:

***

**
*

*
*
*

4:

****

***
*

**
**

**
*
*

*
*
*
*

And so on. (With 5, there are 10 possibilities.)

The question is, can you come up with some kind of formula for the sequence that give the number of possible shapes for each number of stars? The first few numbers, based on the above examples, would be: 1, 2, 3, 5.... I think there is a way to generate the numbers recursively, but I don't know it yet.