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

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, 30, 33, 24, 29, 26, 31, 32, 27, 35, 34, 36, 45, 48, 46, 49, 50, 44, 47, 42, 37, 39, 43, 38, 40, 41, 58, 59, 57, 54, 55, 56, 53, 51, 52, 63, 62, 61, 60, 64, 65, 79, 70, 84, 93

Offset: 0

Author

Antti Karttunen, Jun 23 2003

Keywords

• nonn

Comments

This Catalan bijection reflects the interpretations (pp)-(rr) of Stanley, obtained with the "descending slope mapping" from the Dyck paths encoded by A014486.

Examples

			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:
..................................................
.....___________..................................
....|...|....._.|.................................
....|..||...||.||..................___________....
....|..||...||.||.................|...|...._..|...
....|..||../\|.||..i.e..equal.to..|.|.|.|.|.|.|...
....|./\|./..\/\|.................|.|.|.|.|.|.|...
.../\/..\/......\.................|.|.|.|.|.|.|...
...10110011100100=11492=A014486(250)
Now the Catalan bijection gma086431 gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
.....___________..................................
....|...._..|...|.................................
....|...|.|||..||..................___________....
....|...|.|||..||.................|.._....|...|...
....|../\/\||..||..i.e..equal.to..|.|.|.|.|.|.|...
....|./....\|./\|.................|.|.|.|.|.|.|...
.../\/......\/..\.................|.|.|.|.|.|.|...
...10111010001100=11916=A014486(296)
So we have A086431(250)=296 and A086431(296)=250.
		

Links

• A. Karttunen, Noncrossing Murasaki diagrams obtained via descending slope mapping illustrated up to seven sticks
• A. Karttunen, Gatomorphisms (With the complete Scheme source)
• R. P. Stanley, Exercises on Catalan and Related Numbers (including 66 combinatorial interpretations)
• Index entries for signature-permutations induced by Catalan automorphisms

Crossrefs

a(n) = A057164(A085161(A057164(n))) = A086425(A057164(A086426(n))). Occurs in A073200. Cf. also A086427, A086430.

Number of cycles: A007123. Number of fixed points: A001405. (In range [A014137(n-1)..A014138(n-1)] of this permutation.).