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Signature permutation of a Catalan automorphism: apply *A074679 to the root and recurse down the cdr-spine (the right-hand side edge of a binary tree) as long as the binary tree rotation is possible and if the top-level length (A057515(n)) is odd, then apply *A069770 to the last branch-point.
10

%I #5 Mar 31 2012 13:21:14

%S 0,1,3,2,6,7,8,4,5,15,14,16,17,18,19,20,21,9,10,22,11,12,13,39,40,41,

%T 37,38,43,42,44,45,46,47,48,49,50,52,51,53,54,55,56,57,58,23,24,59,25,

%U 26,27,60,61,62,28,29,63,30,31,32,64,33,34,35,36,113,112,114,115,116

%N Signature permutation of a Catalan automorphism: apply *A074679 to the root and recurse down the cdr-spine (the right-hand side edge of a binary tree) as long as the binary tree rotation is possible and if the top-level length (A057515(n)) is odd, then apply *A069770 to the last branch-point.

%C This automorphism converts lists of even length (1 2 3 4 ... 2n-1 2n) to the form ((1 . 2) (3 . 4) ... (2n-1 . 2n)) and when applied to lists of odd length, like (1 2 3 4 5), i.e. (1 . (2 . (3 . (4 . (5 . ()))))), converts them as ((1 . 2) . ((3 . 4) . (() . 5))).

%H A. Karttunen, <a href="/A130369/b130369.txt">Table of n, a(n) for n = 0..2055</a>

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>

%o (Destructive Scheme implementation of this automorphism, which acts on S-expressions, i.e. list-structures:)

%o (define (*A130369! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (*A069770! s)) (else (*A074679! s) (*A130369! (cdr s)))) s)

%Y Inverse: A130370. a(n) = A074685(A130372(n)) = A130376(A074685(n)). The number of cycles, number of fixed points, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A130377, LEFT(A019590), A130378 and A130379.

%K nonn

%O 0,3

%A _Antti Karttunen_, Jun 05 2007