Mike Stay
Apr24-04, 12:16 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>baez@math.removethis.ucr.andthis.edu (John Baez) wrote in message news:<c1t81n\\$kgm\\$1@glue.ucr.edu>...\n> In article <c1o4rl\\$t4l\\$1@news2.netvision.net.il>,\n> Squark <fiis5d@yahoo.com> wrote:\n>\n> >"John Baez" <baez@math.removethis.ucr.andthis.edu> wrote in message\n> >news:c19sa7\\$at6\\$1@glue.ucr.edu...\n>\n> >> |T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...\n> >>\n> >> Lo and behold! The coefficient of x^n is the number of binary trees\n> >> with n leaves!\n>\n> ... and then I hinted that you could calculate |T| by starting with\n> some initial guess, repeatedly feeding it back into this formula:\n>\n> |T| = x + |T|^2\n>\n> and taking the limit.\n>\n> >A mind boggling thing occured to me: the partition function of rooted planar\n> >binary trees f(x) has the property that f(x) "converges" to infinity if\n> >and only if x lies outside the Mandelbrot set!\n>\n> Wow! You\'re right! So, the Mandelbrot set, the Catalan numbers,\n> and the golden ratio are all related.\n>\n> Cool - I\'ll stick your comment on the website version of "week202".\n\nThis is explored a bit on page 161 of Flajolet and Sedgewick\'s\nAnalytic Combinatorics / Symbolic Combinatorics. (A worthy companion\nto Wilf\'s Generatingfunctionology!)\n\n>\n> Of course, here you are talking about the iterative approach to\n> calculating |T| as a function of x. If we try to calculate it by\n> summing the series, it converges iff |x| < 1/4, since\n>\n> |T|(x) = (1 - sqrt(1 - 4x))/2\n>\n> For further zany relations between the golden ratio, Catalan numbers\n> and chaos, see "week203".\n\n--\nMike Stay\nhttp://www.cs.auckland.ac.nz/~msta039\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>baez@math.removethis.ucr.andthis.edu (John Baez) wrote in message news:<c1t81n$kgm$1@glue.ucr.edu>...
> In article <c1o4rl$t4l$1@news2.netvision.net.il>,
> Squark <fiis5d@yahoo.com> wrote:
>
> >"John Baez" <baez@math.removethis.ucr.andthis.edu> wrote in message
> >news:c19sa7$at6$1@glue.ucr.edu...
>
> >> |T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...
> >>
> >> Lo and behold! The coefficient of x^n is the number of binary trees
> >> with n leaves!
>
> ... and then I hinted that you could calculate |T| by starting with
> some initial guess, repeatedly feeding it back into this formula:
>
> |T| = x + |T|^2
>
> and taking the limit.
>
> >A mind boggling thing occured to me: the partition function of rooted planar
> >binary trees f(x) has the property that f(x) "converges" to infinity if
> >and only if x lies outside the Mandelbrot set!
>
> Wow! You're right! So, the Mandelbrot set, the Catalan numbers,
> and the golden ratio are all related.
>
> Cool - I'll stick your comment on the website version of "week202".
This is explored a bit on page 161 of Flajolet and Sedgewick's
Analytic Combinatorics / Symbolic Combinatorics. (A worthy companion
to Wilf's Generatingfunctionology!)
>
> Of course, here you are talking about the iterative approach to
> calculating |T| as a function of x. If we try to calculate it by
> summing the series, it converges iff |x| < 1/4, since
>
> |T|(x) = (1 - \sqrt(1 - 4x))/2
>
> For further zany relations between the golden ratio, Catalan numbers
> and chaos, see "week203".
--
Mike Stay
http://www.cs.auckland.ac.nz/~msta039
> In article <c1o4rl$t4l$1@news2.netvision.net.il>,
> Squark <fiis5d@yahoo.com> wrote:
>
> >"John Baez" <baez@math.removethis.ucr.andthis.edu> wrote in message
> >news:c19sa7$at6$1@glue.ucr.edu...
>
> >> |T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...
> >>
> >> Lo and behold! The coefficient of x^n is the number of binary trees
> >> with n leaves!
>
> ... and then I hinted that you could calculate |T| by starting with
> some initial guess, repeatedly feeding it back into this formula:
>
> |T| = x + |T|^2
>
> and taking the limit.
>
> >A mind boggling thing occured to me: the partition function of rooted planar
> >binary trees f(x) has the property that f(x) "converges" to infinity if
> >and only if x lies outside the Mandelbrot set!
>
> Wow! You're right! So, the Mandelbrot set, the Catalan numbers,
> and the golden ratio are all related.
>
> Cool - I'll stick your comment on the website version of "week202".
This is explored a bit on page 161 of Flajolet and Sedgewick's
Analytic Combinatorics / Symbolic Combinatorics. (A worthy companion
to Wilf's Generatingfunctionology!)
>
> Of course, here you are talking about the iterative approach to
> calculating |T| as a function of x. If we try to calculate it by
> summing the series, it converges iff |x| < 1/4, since
>
> |T|(x) = (1 - \sqrt(1 - 4x))/2
>
> For further zany relations between the golden ratio, Catalan numbers
> and chaos, see "week203".
--
Mike Stay
http://www.cs.auckland.ac.nz/~msta039