A057164 Self-inverse permutation of natural numbers induced by reflections of the rooted plane trees and mountain ranges encoded by A014486.

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 11, 16, 19, 10, 15, 12, 17, 20, 13, 18, 21, 22, 23, 37, 28, 42, 51, 25, 39, 30, 44, 53, 33, 47, 56, 60, 24, 38, 29, 43, 52, 26, 40, 31, 45, 54, 34, 48, 57, 61, 27, 41, 32, 46, 55, 35, 49, 58, 62, 36, 50, 59, 63, 64, 65, 107, 79, 121, 149, 70

Offset: 0

Author

Antti Karttunen, Aug 18 2000

Keywords

• nonn

Comments

CatalanRankGlobal given in A057117 and the other Maple procedures in A056539.

Composition with A057163 gives Donaghey's Map M (A057505/A057506).

Examples

			a(10)=14 and a(14)=10, A014486[10] = 172 (10101100 in binary), A014486[14] = 202 (11001010 in binary) and these encode the following mountain ranges (and the corresponding rooted plane trees), which are reflections of each other:
...../\___________/\
/\/\/__\_________/__\/\/\
...
...../...........\
..\|/.............\|/
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6917 (first 197 terms from Indranil Ghosh)
• Antti Karttunen, Gatomorphisms (includes the complete Scheme program for computing this sequence).
• Antti Karttunen, C program which implements this Catalan bijection.
• Indranil Ghosh, Python program for computing the sequence.
• Rémy Sigrist, PARI program.
• Index entries for signature-permutations induced by Catalan automorphisms

Crossrefs

A057123(A057163(n)) = A057164(A057123(n)) for all n. Also the car/cdr-flipped conjugate of A069787, i.e., A057164(n) = A057163(A069787(A057163(n))). Fixed terms are given by A061856. Cf. also A057508, A069772.

Row 2 of tables A122287 and A122288.

Programs

Maple

a(n) = CatalanRankGlobal(runcounts2binexp(reverse(binexp2runcounts(A014486[n])))) # i.e., reverse and complement the totally balanced binary sequences

PARI

See Links section.

Formula

a(n) = A080300(A036044(A014486(n))) = A080300(A056539(A014486(n))).