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%I #13 Jan 07 2024 08:16:01
%S 0,1,3,2,7,8,6,4,5,17,18,20,21,22,14,15,16,9,10,19,11,12,13,45,46,48,
%T 49,50,54,55,57,58,59,61,62,63,64,37,38,39,40,41,42,43,44,23,24,47,25,
%U 26,27,51,52,53,28,29,56,30,31,32,60,33,34,35,36,129,130,132,133,134
%N Self-inverse signature permutation of a Catalan bijection: row 183 of A089840.
%C This bijection of binary trees swaps the left and right subtree of a binary tree, but ONLY if either of them is empty. If both the left and right hand side tree is nonempty, fixes the tree.
%C .A...B.C...D.......A...B.C...D.....
%C ..\./...\./.........\./...\./........................
%C ...x.....x...--->....x.....x.......A...B.......B...A.
%C ....\.../.............\.../.........\./..--->...\./..
%C ......x.................x............x...........x...
%C ..............................(where either A or B is (), a leaf)
%C This automorphism demonstrates that not every clause in clause-representations of A089840 is equal to some (minimally represented) element of Thompson's group V.
%H A. Karttunen, <a href="/A154126/b154126.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://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 (*A154126! s) (if (and (pair? s) (or (not (pair? (car s))) (not (pair? (cdr s))))) (*A069770! s)) s)
%Y Inverse: A154126. a(n) = A069770(A154125(n)) = A154125(A069770(n)).
%K nonn
%O 0,3
%A _Antti Karttunen_, Jan 06 2009
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