tag:blogger.com,1999:blog-264226589944705290.post5196659339439717331..comments2022-05-19T07:55:48.367-07:00Comments on God Plays Dice: Splitting permutations of [n] into two classesMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-264226589944705290.post-37694683287689396532008-02-11T10:51:00.000-08:002008-02-11T10:51:00.000-08:00In general, I like generating function arguments t...In general, I like generating function arguments too. And, I do like this one in particular.<BR/><BR/>A small typo: the second box containing the infinite series should have (3!z^4)/4! as its fourth term.<BR/><BR/>(The second comment btw is mine.)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-27359534398723736322008-02-11T03:18:00.000-08:002008-02-11T03:18:00.000-08:00I actually liked the generating function argument ...I actually liked the generating function argument more. I haven't seen that sort of combinatorics for years, and it was one of my favorite tools when I was an undergrad math major.CarlBrannenhttps://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-76374222382115922782008-02-10T22:26:00.000-08:002008-02-10T22:26:00.000-08:00I think this result is a common one in an undergra...I think this result is a common one in an undergraduate abstract algebra course, and the bijective proof provided by Jeremy is the usual one. Fraleigh's book does contain such a proof.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-9865792683373676962008-02-10T12:21:00.000-08:002008-02-10T12:21:00.000-08:00A bijective proof is quite straightforward: if s i...A bijective proof is quite straightforward: if s is a permutation and t is a 2-cycle, then t.s has either one more cycle than t (if t interchanges elements in the same cycle of s, because it splits that cycle into two) or one less (if t interchanges elements in different cycles of s, because it joins those cycles together). Ergo, multiplication by t is a bijection that reverses the parity of the number of cycles.<BR/><BR/>Jeremy HentyAnonymousnoreply@blogger.com