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%I #16 Jan 07 2024 08:25:20
%S 0,1,2,3,5,4,6,7,8,11,12,13,9,10,15,14,16,17,18,19,20,21,22,28,29,30,
%T 31,32,33,34,35,23,24,36,25,26,27,39,40,41,37,38,43,42,44,45,46,47,48,
%U 49,50,52,51,53,54,55,56,57,58,59,60,61,62,63,64,79,80,81,82,83,84,85
%N Signature permutation of a Catalan bijection: row 3655 of A089840.
%C This bijection of binary trees can be obtained by applying bijection *A074679 to the right hand side subtree and leaving the left hand side subtree intact:
%C ....C...D.......B...C
%C .....\./.........\./
%C ..B...x....-->....x...D.................B..().........()..B..
%C ...\./.............\./...................\./....-->....\./...
%C A...x...........A...x.................A...x.........A...x....
%C .\./.............\./...................\./...........\./.....
%C ..x...............x.....................x.............x......
%C .............................................................
%C Note that the first clause corresponds to generator B of Thompson's groups F, T and V, while *A074679's first clause corresponds to generator A and furthermore, *A089851 corresponds to generator C and *A072796 to generator pi_0 of Thompson's group V. (To be checked: can Thompson's V be embedded in A089840 by using these or some other suitably chosen generators?)
%C Comment to above: I think now that it is a misplaced hope to embed V in A089840. Instead, it is more probable that Thompson's V is isomorphic to the quotient group A089840/N, where N is a subgroup of A089840 which includes identity (*A001477) and any other bijection (e.g. *A154126) that fixes all large enough trees. For more details, see my "On the connection of A089840 with ..." page. - _Antti Karttunen_, Aug 23 2012
%H A. Karttunen, <a href="/A154121/b154121.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. Karttunen, <a href="http://oeis.org/wiki/User:Antti_Karttunen/Speculations/On_the_connection_of_A089840_with_Thompsons_groups">On the connection of A089840 with Thompson's groups</a>, speculation and conjectures.
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%o (Destructive version of this automorphism in Scheme:) (define (*A154121! s) (if (pair? s) (*A074679! (cdr s))) s)
%Y Inverse: A154122. a(n) = A069770(A089865(A069770(n))). Cf. A154123, A154126.
%K nonn
%O 0,3
%A _Antti Karttunen_, Jan 06 2009
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