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A125976
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Signature-permutation of Kreweras' 1970 involution on Dyck paths.
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13
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0, 1, 3, 2, 8, 6, 5, 7, 4, 22, 19, 15, 20, 14, 13, 11, 18, 21, 16, 10, 12, 17, 9, 64, 60, 52, 61, 51, 41, 39, 55, 62, 53, 38, 40, 54, 37, 36, 33, 29, 34, 28, 50, 47, 59, 63, 56, 43, 48, 57, 42, 27, 25, 32, 35, 30, 46, 49, 58, 44, 24, 26, 31, 45, 23, 196, 191, 178, 192, 177
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Lalanne shows in the 1992 paper that this automorphism preserves the sum of peak heights, i.e. that A126302(a(n)) = A126302(n) for all n. Furthermore, he also shows that A126306(a(n)) = A057514(n)-1 and likewise, that A057514(a(n)) = A126306(n)+1, for all n >= 1.
Like A069772, this involution keeps symmetric Dyck paths symmetric, but not necessarily same.
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REFERENCES
| G. Kreweras, Sur les eventails de segments, Cahiers du Bureau Universitaire de Recherche Operationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
J.-C. Lalanne, Une Involution sur les Chemins de Dyck, European J. Combin. 13 (1992), no. 6, 477-487.
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LINKS
| A. Karttunen, Table of n, a(n) for n = 0..2055
Index entries for signature-permutations of Catalan automorphisms
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CROSSREFS
| a(n) = A080300(A125974(A014486(n))). The number of cycles and fixed points in range [A014137(n-1)..A014138(n-1)] of this involution seem to be given by A007595 and the "aerated" Catalans [1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, ...], thus this is probably a conjugate of A069770 (as well as of A057163). Compositions and conjugations with other automorphisms: A125977-A125979, A125980, A126290.
Sequence in context: A191537 A132827 A126315 * A071654 A072657 A098163
Adjacent sequences: A125973 A125974 A125975 * A125977 A125978 A125979
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KEYWORD
| nonn
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AUTHOR
| Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jan 02 2007
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