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%I #19 Sep 09 2017 19:47:37
%S 0,1,3,2,8,6,7,4,5,22,19,20,14,15,21,16,17,9,10,18,11,12,13,64,60,61,
%T 51,52,62,53,54,37,38,55,39,40,41,63,56,57,42,43,58,44,45,23,24,46,25,
%U 26,27,59,47,48,28,29,49,30,31,32,50,33,34,35,36,196,191,192,177,178
%N Signature-permutation of a Catalan Automorphism: rotate one step clockwise the triangulations of polygons encoded by A014486.
%C This is a permutation of natural numbers induced when Euler's triangulation of convex polygons, encoded by the sequence A014486 in a straightforward way (via binary trees, cf. the illustration of the rotation of a triangulated pentagon, given in the Links section) are rotated clockwise.
%C In A057161 and A057162, the cycles between A014138(n-1)-th and A014138(n)-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A001683(n+2) equivalence classes of flexagons (or unlabeled plane boron trees), thus the latter sequence can be counted with the Maple procedure A057162_CycleCounts given below. Cf. also the comments in A057161.
%H A. Karttunen, <a href="/A057162/b057162.txt">Table of n, a(n) for n = 0..2055</a>
%H A. Karttunen, <a href="/A057161/a057161.svg">Illustration of how the five triangulations of a pentagon will rotate, and the corresponding changes it induces in the binary trees</a>
%H A. Karttunen, <a href="http://web.archive.org/web/20121004142217/http://ndirty.cute.fi/~karttu/matikka/Nekomorphisms/CatBijections.pdf">Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft)</a>, pp. 51-54.
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%F As a composition of related permutations:
%F a(n) = A069768(A057508(n)).
%F a(n) = A057163(A057161(A057163(n))).
%F a(n) = A057164(A057503(A057164(n))). [For the proof, see pp. 53-54 in the "Introductory survey ..." draft, eq. 143.]
%p a(n) = CatalanRankGlobal(RotateTriangularizationR(A014486[n]))
%p RotateTriangularizationR := n -> ReflectBinTree(RotateTriangularization(ReflectBinTree(n)));
%p with(group); A057162_CycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateTriangularization(CatalanUnrank(n,r)))]; od; a := [op(a),(`if`((n < 2),1,nops(convert(b,'disjcyc'))))]; od; RETURN(a); end;
%p # See also the code in A057161.
%o (Scheme functions implementing this automorphism on S-expressions, three different variants):
%o (define (*A057162 bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (car lt)))))))
%o (define (*A057162 s) (fold-right (lambda (x y) (*A057163 (append (*A057163 y) (list (*A057163 x))))) (quote ()) s))
%o (define (*A057162! s) (*A057508! s) (*A069768! s) s)
%Y Inverse: A057161.
%Y Also, an "ENIPS"-transform of A069773, and thus occurs as row 17 of A130402.
%Y Other related permutations: A057163, A057164, A057501, A057503, A057505.
%Y Cf. A001683 (cycle counts), A057544 (max cycle lengths).
%K nonn
%O 0,3
%A _Antti Karttunen_, Aug 18 2000; entry revised Jun 06 2014
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