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%I #10 Jan 07 2024 08:44:51
%S 0,1,2,3,5,6,4,7,8,11,12,13,16,19,15,14,9,17,18,10,20,21,22,28,29,30,
%T 31,32,33,34,35,44,47,36,53,56,60,39,40,41,42,51,43,37,23,45,46,24,48,
%U 49,50,52,38,25,54,55,26,57,58,59,27,61,62,63,64,79,80,81,82,83,84,85
%N Signature permutation of a Catalan bijection: row 3656 of A089840.
%C This bijection of binary trees is obtained in the following way. See also comments at A154122.
%C ....C...D.......B...C
%C .....\./.........\./
%C ..B...x....-->....x...D.................B..().........()..A..
%C ...\./.............\./...................\./....-->....\./...
%C A...x...........A...x.................A...x.........B...x....
%C .\./.............\./...................\./...........\./.....
%C ..x...............x.....................x.............x......
%C .............................................................
%C That is, we do (a . (b . (c . d))) -> (a . ((b . c) . d))
%C or (a . (b . ())) --> (b . (() . a)) if the former is not possible.
%C Note that the first clause corresponds to generator B of Thompson's groups F, T and V. See further comments at A154121.
%H A. Karttunen, <a href="/A154123/b154123.txt">Table of n, a(n) for n = 0..2055</a>
%H J. W. Cannon, W. J. Floyd, and W. R. Parry, <a href="https://web.archive.org/web/20110605062246/http://www.geom.uiuc.edu/docs/preprints/lib/GCG63/thompson.ps">Notes on Richard Thompson's Groups F and T</a>
%H J. W. Cannon, W. J. Floyd, and W. R. Parry, <a href="https://doi.org/10.5169/seals-87877">Introductory notes on Richard Thompson's groups</a>, L'Enseignement Mathématique, Vol. 42 (1996), pp. 215-256.
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o (Destructive version of this automorphism in Scheme:) (define (*A154123! s) (if (and (pair? s) (pair? (cdr s)) (pair? (cddr s))) (*A074679! (cdr s)) (*A089851! s)) s)
%Y Inverse: A154124. Cf. A154121.
%K nonn
%O 0,3
%A _Antti Karttunen_, Jan 06 2009
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