A057162 Signature-permutation of a Catalan Automorphism: rotate one step clockwise the triangulations of polygons encoded by A014486.

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, 51, 52, 62, 53, 54, 37, 38, 55, 39, 40, 41, 63, 56, 57, 42, 43, 58, 44, 45, 23, 24, 46, 25, 26, 27, 59, 47, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 196, 191, 192, 177, 178

Offset: 0

Author

Antti Karttunen, Aug 18 2000; entry revised Jun 06 2014

Keywords

• nonn

Comments

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.

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.

Links

• A. Karttunen, Table of n, a(n) for n = 0..2055
• A. Karttunen, Illustration of how the five triangulations of a pentagon will rotate, and the corresponding changes it induces in the binary trees
• A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft), pp. 51-54.
• Index entries for signature-permutations of Catalan automorphisms

Crossrefs

Inverse: A057161.

Also, an "ENIPS"-transform of A069773, and thus occurs as row 17 of A130402.

Other related permutations: A057163, A057164, A057501, A057503, A057505.

Cf. A001683 (cycle counts), A057544 (max cycle lengths).

Programs

Maple

a(n) = CatalanRankGlobal(RotateTriangularizationR(A014486[n]))
RotateTriangularizationR := n -> ReflectBinTree(RotateTriangularization(ReflectBinTree(n)));
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;
# See also the code in A057161.

Formula

As a composition of related permutations:

a(n) = A069768(A057508(n)).

a(n) = A057163(A057161(A057163(n))).

a(n) = A057164(A057503(A057164(n))). [For the proof, see pp. 53-54 in the "Introductory survey ..." draft, eq. 143.]