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A072796 Self-inverse permutation of natural numbers induced by Catalan Automorphism *A072796 acting on the parenthesizations encoded by A014486. 50
0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 14, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)

...B...C...........A...C

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

.A...x....-->....B...x.................A..().........A...()..

..\./.............\./...................\./....-->....\./...

...x...............x.....................x.............x....

(a . (b . c)) -> (b . (a . c)) ______ (a . ()) ---> (a . ())

In terms of general trees, it swaps the two leftmost branches of the tree if its degree (i.e. of the root) > 1 and keeps the tree intact if it is planted (root's degree = 1).

Note that the first clause corresponds to what is called "generator pi_0" in Thompson's group V. (See also A074679, A089851 and A154121 for other related generators).

Look at the example in A069770 to see how this will produce the given sequence of integers.

REFERENCES

J. W. Cannon, W. J. Floyd and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathematique, Vol. 42 (1996), pp. 215--256.

LINKS

A. Karttunen, Table of n, a(n) for n = 0..196

J. W. Cannon, W. J. Floyd and W. R. Parry, Notes on Richard Thompson's Groups F and T

A. Karttunen, Catalan Automorphisms

PROG

(Scheme function implementing this automorphism on list-structures, both the constructive (*A072796) and destructive (*A072796!) variant given:)

(define (*A072796 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (cons (cadr s) (cons (car s) (cddr s))))))

(define (*A072796! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (swap! s) (robr! s) (swap! (cdr s)) s)))

(define (robr! s) (let ((ex-cdr (cdr s))) (set-cdr! s (caar s)) (set-car! (car s) ex-cdr) (swap! (car s)) (swap! s) s))

(define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))

CROSSREFS

Row 2 of A089840. Row 3613 of A122203 and row 3617 of A122204.

Fixed point counts and cycle counts are given in A073190 and A073191.

Cf. A073190, A072797, A129608.

Sequence in context: A058812 A320454 A320455 * A130374 A122363 A122364

Adjacent sequences:  A072793 A072794 A072795 * A072797 A072798 A072799

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 12 2002

STATUS

approved

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)