<< I’m not sure who deserves credit for this observation; perhaps one of my readers knows. >> Try Raney's lemma --- see eg. http://www.cs.tau.ac.il/~nachum/papers/CL.pdf The number of smileys equals the Catalan number C(n) , the number of ways to arrange n pairs of brackets legally: see http://oeis.org/A000108 I've long been interested in these questions myself, so hardly represent a typical reader. But I did feel that this article moved rather too slowly for comfort. It might be worth taking a quick pass through while asking how much chatty padding could be ruthlessly expunged without damage to your argument. Fred Lunnon On 8/10/16, James Propp <jamespropp@gmail.com> wrote:

I started writing a new draft titled "Bertrand's Ballot Problem" and would love to get your feedback. I plan on publishing it on the 17th. It's still quite rough (and I haven't made the figures yet), but I thought I'd share it now, given that August 17th is only a week away. I'll probably update it now and then during the week ahead.

I'm hoping to learn Omnigraffle by using it to prepare the eight illustrations. Are any of you Omnigraffle users, and hence potentially available to offer me assistance if I have an "Oh Mister Wizard! Mister Wizard!" moment?

Suggestions for references, and comments of all kinds, are welcome.

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Please leave your feedback here: https://mathenchant.wordpress.com?p=879&shareadraft=57a549f09f22d

Title: Bertrand's Ballot Problem Beginning: An early lead in an election doesn't necessarily mean much. Remember Republican presidential candidate Rudy Giuliani? Back in 2008, the outspoken New Yorker polled so well that some pundits predicted he'd win the nomination, but his lead fizzled out before the convention. The same thing has happened... Read more: https://mathenchant.wordpress.com?p=879&shareadraft=57a549f0 9f22d

Thanks, jamespropp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun