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A086431 Involution of natural numbers induced by the Catalan bijection gma086431 acting on symbolless S-expressions encoded by A014486/A063171.

%I A086431 #9 Oct 15 2015 10:55:25
%S A086431 0,1,2,3,4,5,7,6,8,9,11,10,12,13,17,18,16,14,15,21,20,19,22,23,28,25,
%T A086431 30,33,24,29,26,31,32,27,35,34,36,45,48,46,49,50,44,47,42,37,39,43,38,
%U A086431 40,41,58,59,57,54,55,56,53,51,52,63,62,61,60,64,65,79,70,84,93
%N A086431 Involution of natural numbers induced by the Catalan bijection gma086431 acting on symbolless S-expressions encoded by A014486/A063171.
%C A086431 This Catalan bijection reflects the interpretations (pp)-(rr) of Stanley, obtained with the "descending slope mapping" from the Dyck paths encoded by A014486.
%H A086431 A. Karttunen, <a href="/A014486/a014486.ps.gz">Noncrossing Murasaki diagrams obtained via descending slope mapping illustrated up to seven sticks</a>
%H A086431 A. Karttunen, <a href="http://www.iki.fi/~kartturi/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</a> (With the complete Scheme source)
%H A086431 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/ec/catalan.pdf">Exercises on Catalan and Related Numbers</a> (including 66 combinatorial interpretations)
%H A086431 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%e A086431 Map the Dyck paths (Stanley's interpretation (i)) to noncrossing Murasaki-diagrams (Stanley's interpretation (rr)) by drawing a vertical line above each descending slope \ and connect those vertical lines that originate from the same height without any lower valleys between, as in illustration below:
%e A086431 ..................................................
%e A086431 .....___________..................................
%e A086431 ....|...|....._.|.................................
%e A086431 ....|..||...||.||..................___________....
%e A086431 ....|..||...||.||.................|...|...._..|...
%e A086431 ....|..||../\|.||..i.e..equal.to..|.|.|.|.|.|.|...
%e A086431 ....|./\|./..\/\|.................|.|.|.|.|.|.|...
%e A086431 .../\/..\/......\.................|.|.|.|.|.|.|...
%e A086431 ...10110011100100=11492=A014486(250)
%e A086431 Now the Catalan bijection gma086431 gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
%e A086431 .....___________..................................
%e A086431 ....|...._..|...|.................................
%e A086431 ....|...|.|||..||..................___________....
%e A086431 ....|...|.|||..||.................|.._....|...|...
%e A086431 ....|../\/\||..||..i.e..equal.to..|.|.|.|.|.|.|...
%e A086431 ....|./....\|./\|.................|.|.|.|.|.|.|...
%e A086431 .../\/......\/..\.................|.|.|.|.|.|.|...
%e A086431 ...10111010001100=11916=A014486(296)
%e A086431 So we have A086431(250)=296 and A086431(296)=250.
%Y A086431 a(n) = A057164(A085161(A057164(n))) = A086425(A057164(A086426(n))). Occurs in A073200. Cf. also A086427, A086430.
%Y A086431 Number of cycles: A007123. Number of fixed points: A001405. (In range [A014137(n-1)..A014138(n-1)] of this permutation.).
%K A086431 nonn
%O A086431 0,3
%A A086431 _Antti Karttunen_, Jun 23 2003