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Search: id:A085161
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| A085161 |
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Involution of natural numbers induced by the gatomorphism gma085161 acting on symbolless S-expressions encoded by A014486/A063171. |
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+0
12
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| 0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 17, 14, 12, 21, 11, 20, 16, 10, 18, 19, 15, 13, 22, 23, 45, 37, 31, 58, 28, 54, 42, 26, 49, 51, 40, 35, 63, 25, 48, 39, 34, 62, 30, 57, 44, 24, 46, 56, 38, 32, 59, 33, 61, 53, 29, 55, 47, 43, 27, 50, 60, 52, 41, 36, 64, 65, 129, 107, 87, 170
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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This gatomorphism reflects the interpretations (pp)-(rr) of Stanley, obtained from the Dyck paths with the "rising slope mapping" illustrated on the example lines.
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LINKS
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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
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EXAMPLE
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Map the Dyck paths (Stanley's interpretation (i)) to noncrossing Murasaki-diagrams (Stanley's interpretation (rr)) by drawing a vertical line above each rising 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 gatomorphism gma085161 gives the parenthesization such that the corresponding Murasaki-diagram is a reflection of the original one:
....___.._____....................................
...|...||...|.|...................................
...||..|||..|.|....................___..._____....
...||..|||..|.|...................|...|.|...|.|...
...||..||/\.|.|....i.e..equal.to..|.|.|.|.|.|.|...
...|/\.|/..\/\/\..................|.|.|.|.|.|.|...
.../..\/........\.................|.|.|.|.|.|.|...
...11001110010100=13204=A014486(360)..............
...(())((())()()).................................
So we have A085161(250)=360 and A085161(360)=250.
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PROGRAM
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(Scheme function implementing this automorphism on list-structures:)
(define (gma085161 s) (cond ((null? s) s) (else (let ((u (reverse s))) (app-to-xrt (gma085161 (car u)) (append (map gma085161 (cdr u)) (list (list))))))))
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CROSSREFS
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a(n) = A085163(A057508(n)) = A074684(A057164(A074683(n))). Occurs in A073200. Cf. also A085159, A085160, A085162, A085175. Alternative mappings illustrated in A086431 & A085169. Scheme-function app-to-xrt given in A085203.
Number of cycles: A007123. Number of fixed points: A001405. (In range [A014137(n-1)..A014138(n-1)] of this permutation.).
Sequence in context: A092842 A072028 A072026 this_sequence A085162 A106453 A122199
Adjacent sequences: A085158 A085159 A085160 this_sequence A085162 A085163 A085164
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen (Firstname.Surname(AT)iki.fi), Jun 23 2003
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