cp's OEIS Frontend

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

%I A125976 #18 Mar 22 2016 09:38:39
%S A125976 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,
%T A125976 61,51,41,39,55,62,53,38,40,54,37,36,33,29,34,28,50,47,59,63,56,43,48,
%U A125976 57,42,27,25,32,35,30,46,49,58,44,24,26,31,45,23,196,191,178,192,177
%N A125976 Signature-permutation of Kreweras' 1970 involution on Dyck paths.
%C A125976 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.
%C A125976 Like A069772, this involution keeps symmetric Dyck paths symmetric, but not necessarily same.
%C A125976 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).
%H A125976 A. Karttunen, <a href="/A125976/b125976.txt">Table of n, a(n) for n = 0..2055</a>
%H A125976 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1970__15__3_0">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
%H A125976 J.-C. Lalanne, <a href="http://dx.doi.org/10.1016/0195-6698(92)90006-L">Une Involution sur les Chemins de Dyck</a>, European J. Combin. 13 (1992), no. 6, 477-487.
%H A125976 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations of Catalan automorphisms</a>
%F A125976 a(n) = A080300(A125974(A014486(n))).
%Y A125976 Cf. A080300, A125974, A014486.
%Y A125976 Cf. A057163, A069770, A007595, A014137, A014138.
%Y A125976 Compositions and conjugations with other automorphisms: A125977-A125979, A125980, A126290.
%K A125976 nonn
%O A125976 0,3
%A A125976 _Antti Karttunen_, Jan 02 2007