This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085169 #10 Oct 17 2015 08:36:24 %S A085169 0,1,2,3,4,5,6,8,7,9,10,11,13,12,14,15,19,22,21,16,18,17,20,23,24,25, %T A085169 27,26,28,29,33,36,35,30,32,31,34,37,38,39,41,40,51,52,60,64,63,56,59, %U A085169 58,62,42,43,47,50,49,44,46,45,48,53,55,54,61,57,65,66,67,69,68 %N A085169 Permutation of natural numbers induced by the Catalan bijection gma085169 acting on symbolless S-expressions encoded by A014486/A063171. %C A085169 A parenthesization is fixed by the Catalan bijections A085169/A085170 if and only if no other elements than () and (()) occur at its top-level: (); ()(),(()); ()()(),()(()),(())(); ()()()(),()()(()),()(())(),(())()(),(())(()); ... There is a simple bijection between these and Zeckendorf-expansions, explaining why Fibonacci numbers gives the number of fixed points of this permutation. %C A085169 In addition to "rising slope" and "descending slope" mappings from Dyck paths to noncrossing Murasaki-diagrams as illustrated in A085161 and A086431 there is also a mapping where we insert a vertical stick after every second parenthesis and connect those that are on the same level without any intermediate points below. This Catalan bijection converts between these two mappings. See the illustration at example lines. %H A085169 A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</a> (With the complete Scheme source) %H A085169 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a> %e A085169 ......................... %e A085169 ..._____....________..... %e A085169 ..|.....|..|.....|..|.... %e A085169 ..|..|..|..|..|..|..|.... %e A085169 ..|..|..|..|..|..|..|.... %e A085169 ..|..|..|..|..|..|..|.... %e A085169 ..|..|..|..|..|..|..|.... %e A085169 ..1((2))3((4((5))6()7)).. %e A085169 ...(())(((())()))........ %e A085169 ...11001111001000=13256=A014486(368) %e A085169 To obtain the same Murasaki diagram using the "rising slope mapping" illustrated in A085161, we should use the following Dyck path, encoded by 360th binary string in A014486/A063171: %e A085169 ....___.._____........... %e A085169 ...|...||...|.|.......... %e A085169 ...||..|||..|.|.......... %e A085169 ...||..|||..|.|.......... %e A085169 ...||..||/\.|.|.......... %e A085169 ...|/\.|/..\/\/\......... %e A085169 .../..\/........\........ %e A085169 ...11001110010100=13204=A014486(360) %e A085169 So we have A085169(368)=360 and A085170(360)=368. %o A085169 (Scheme function implementing this automorphism on list-structures:) %o A085169 (define (gma085169 s) (letrec ((evenlev (lambda (s) (cond ((not (pair? s)) s) (else (cons (oddlev (car s)) (evenlev (cdr s))))))) (oddlev (lambda (s) (cond ((not (pair? s)) s) (else (append (evenlev (car s)) (list (oddlev (cdr s))))))))) (evenlev s))) %Y A085169 Inverse: A085170. a(n) = A086433(A082853(n))+A082852(n). A074684 = A083925(A085169(A057548(n))). Cf. also A085159, A085160, A085175. %Y A085169 Number of cycles: A086585. Number of fixed points: A000045. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation). %K A085169 nonn %O A085169 0,3 %A A085169 _Antti Karttunen_, Jun 23 2003