A126313 Signature-permutation of a Catalan automorphism: composition of A069772 and A125976.
0, 1, 3, 2, 8, 5, 6, 4, 7, 22, 13, 15, 12, 14, 19, 21, 16, 11, 18, 10, 20, 17, 9, 64, 36, 41, 35, 40, 52, 53, 38, 34, 39, 55, 51, 37, 54, 60, 63, 32, 62, 31, 56, 59, 47, 33, 50, 27, 58, 49, 26, 43, 44, 29, 61, 30, 24, 57, 48, 25, 46, 42, 28, 23, 45, 196, 106, 120, 105, 119
Offset: 0
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Inverse: A126314. a(n) = A069772(A125976(n)) = A126290(A069772(n)) = A126315(A057164(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 A127277, A127278, A127279 and A127280. The fixed points are given by A127306. Note the curiosity: this automorphism partitions the A000108(8) = 1430 Catalan structures of size eight (e.g. Dyck paths of length 16) into 79 equivalence classes, of which the largest contains 79 members.
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