A057164 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

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

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

FORMULA

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

EXAMPLE

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:

...../\___________/\

/\/\/__\_________/__\/\/\

...

...../...........\

..\|/.............\|/

MAPLE

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

PROG

(Scheme function implementing this automorphism on list-structures:) (define (DeepRev lista) (cond ((not (pair? lista)) lista) ((null? (cdr lista)) (cons (DeepRev (car lista)) (list))) (else (append (DeepRev (cdr lista)) (DeepRev (cons (car lista) (list)))))))

(PARI) See Links section.

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.

Sequence in context: A073285 A057512 A057508 * A085175 A130111 A104182

Adjacent sequences: A057161 A057162 A057163 * A057165 A057166 A057167

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 18 2000

STATUS

approved