A125976 Signature-permutation of Kreweras' 1970 involution on Dyck paths.

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

Offset: 0

Author

Antti Karttunen, Jan 02 2007

Keywords

• nonn

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.

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" Catalan numbers [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).

Links

• A. Karttunen, Table of n, a(n) for n = 0..2055
• G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, 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.
• Index entries for signature-permutations of Catalan automorphisms

Crossrefs

Cf. A080300, A125974, A014486.

Cf. A057163, A069770, A007595, A014137, A014138.

Compositions and conjugations with other automorphisms: A125977-A125979, A125980, A126290.

Formula

a(n) = A080300(A125974(A014486(n))).