User li zhou - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:52:58Z http://mathoverflow.net/feeds/user/11299 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74908/generalizing-the-catalan-number-enumerative-combinatorics Generalizing the Catalan number (enumerative combinatorics) Li Zhou 2011-09-08T17:23:41Z 2011-09-08T23:18:58Z <p>Consider any sequence consisting of n A's and n B's so that in any of its initial partial segments, the number of B's never exceed the number of A's. It is well known that the number of such sequences is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$.</p> <p>Now consider sequences consisting of n A's, n B's, and n C's, so that in any initial segments, the number of B's never exceed the number of A's AND the number of C's never exceed the number of B's. Is the number of such sequences known? </p> http://mathoverflow.net/questions/43820/extremely-messy-proofs/58602#58602 Answer by Li Zhou for Extremely messy proofs Li Zhou 2011-03-16T03:01:56Z 2011-03-16T03:01:56Z <p>We should stop crediting "the nice proof" to Niven. It rightfully belongs to Hermite. Niven's proof is a slight "variation" of Hermite's proof. The integral Niven used (a simple change of variable from Hermite's integral) is closely related to Lambert's continued fraction. Read <a href="http://vixra.org/pdf/1011.0077v2.pdf" rel="nofollow">my preprint</a> and the references therein.</p> http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes/48230#48230 Answer by Li Zhou for Proof that pi is transcendental that doesn't use the infinitude of primes Li Zhou 2010-12-03T23:17:36Z 2010-12-03T23:17:36Z <p>My feeling is similar to Barry's: the infinitude of primes may not be necessary. For example, in Chapter 2 of Niven's <em>Irrational Numbers</em>, he also used the infinitude of primes to prove that cos(r) is irrational for nonzero rational r. But our recent proof (<em>Monthly, April/2010, 360-362</em>, mentioned by Will Jagy earlier) does not need this at all. By the way, our proof (half-page long) can replace more or less the entire Chapter 2 of Niven's book (except the transcendence of e).</p> http://mathoverflow.net/questions/43820/extremely-messy-proofs/58602#58602 Comment by Li Zhou Li Zhou 2011-03-16T04:37:42Z 2011-03-16T04:37:42Z Yes, Gerry. I meant to comment on your entry about Niven's proof of the irrationality of pi, but mistakenly started a new answer. http://mathoverflow.net/questions/21367/proof-that-pi-is-transcendental-that-doesnt-use-the-infinitude-of-primes/48230#48230 Comment by Li Zhou Li Zhou 2010-12-04T16:20:36Z 2010-12-04T16:20:36Z I'm now more convinced that the infinitude of primes is not essential. It's used in the proof of the transcendence of pi for very similar purposes as in the proof of the irrationality of cos(r) by Niven, thus may be replaced by recurrences as in our Monthly paper (also available at <a href="http://arxiv.org/abs/0911.1933" rel="nofollow">arxiv.org/abs/0911.1933</a>). The recurrences, however, will be very very messy.