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Today is my mother’s birthday. It was quite a good day I guess. I was awakened at around 10:30 and told we were going to breakfast in a few minutes, so I got up quickly to prepare for the first excursion of the day. Some time after we returned, I decided it would be a good idea to get my mother a birthday present, so I walked down to Borders, a mile from my house, and got her a book she had requested. Of course, in the meantime, I wasn’t there when Guanxiong came by to my house to pick up science bowl material, so he sent me annoyed IMs saying that it would have been nice had I actually been there when he came by. Guanxiong, if you’re reading this, sorry about that! Anyway, after that I got some more scripts to proofread for the Intermediate Counting class at AoPS. Yay. It’s always fun reading through them. Perhaps it’s time for me to learn a combinatorial proof that the nth Catalan number is (2n choose n)/(n+1) rather than relying on generating functions to save me all the time. Come to think of it, I’m not really sure if I can prove it with generating functions. Anyway, after that, we went to dinner. I can’t remember having so much good food in one day before. Well, I’m not going to complain about being spoiled from time to time. Well, now I’m back to the computer, and I should either be learning about Greek city-states or reading AoPS scripts. Those both sound like better options than randomly surfing the web, don’t they? In that case, I’ll go through the forum once more to see if there’s anything new and then do something more productive.


About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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6 Responses to Next Post

  1. z9r4c3 says:

    hehe! well, happy belated birthday to your mum and all! *turns away* *whirls back* You have Science Bowl material?! Where do you find this stuff? And am I allowed to give the ARML practice people my own problems instead?

    • Simon says:

      You have to tell me which problems before I allow or disallow that. If they come from your father, I’ll probably disallow that…

  2. rippledance says:

    *bounce* Does the SB material come from the Tennessee website / the official site? If so I have that already…if not, where did you get it from?

    • Simon says:

      I’ll give you the cop-out answer only right now. I got it from the (former) teacher at my high school (who has now retired) who coached the science bowl team. Of course, I know exactly where it really came from, but I should ask permission before I disclose that information.

  3. aknoln says:

    Uh I think you can just show that by strong induction. Either the path touches the x axis or it doesn’t. There are C_n-2 ways for it to not, and you can sum all the ones for which it does based on where it first touches the x axis.

    • Simon says:

      That’s true. You can prove it with the catalan recursion C_n=C_0*C_(n-1)+C_1*C_(n-2)+…+C_(n-1)*C_0. But I was looking for a nice combinatorial proof. Well, I got one now, and it’s really clever.

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