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A154121 Signature permutation of a Catalan bijection: row 3655 of A089840. 8
0, 1, 2, 3, 5, 4, 6, 7, 8, 11, 12, 13, 9, 10, 15, 14, 16, 17, 18, 19, 20, 21, 22, 28, 29, 30, 31, 32, 33, 34, 35, 23, 24, 36, 25, 26, 27, 39, 40, 41, 37, 38, 43, 42, 44, 45, 46, 47, 48, 49, 50, 52, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 79, 80, 81, 82, 83, 84, 85 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This bijection of binary trees can be obtained by applying bijection *A074679 to the right hand side subtree and leaving the left hand side subtree intact:

....C...D.......B...C

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

..B...x....-->....x...D.................B..().........()..B..

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

A...x...........A...x.................A...x.........A...x....

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

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

.............................................................

Note that the first clause corresponds to generator B of Thompson's groups F, T and V, while *A074679's first clause corresponds to generator A and furthermore, *A089851 corresponds to generator C and *A072796 to generator pi_0 of Thompson's group V. (To be checked: can Thompson's V be embedded in A089840 by using these or some other suitably chosen generators?)

Comment to above: I think now that it is a misplaced hope to embed V in A089840. Instead, it is more probable that Thompson's V is isomorphic to the quotient group A089840/N, where N is a subgroup of A089840 which includes identity (*A001477) and any other bijection (e.g. *A154126) that fixes all large enough trees. For more details, see my "On the connection of A089840 with ..." page. - Antti Karttunen, Aug 23 2012

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..2055

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

A. Karttunen, On the connection of A089840 with Thompson's groups, speculation and conjectures.

Index entries for signature-permutations of Catalan automorphisms

PROG

(Destructive version of this automorphism in Scheme:) (define (*A154121! s) (if (pair? s) (*A074679! (cdr s))) s)

CROSSREFS

Inverse: A154122. a(n) = A069770(A089865(A069770(n))). Cf. A154123, A154126.

Sequence in context: A060119 A060126 A216250 * A130375 A154122 A089850

Adjacent sequences:  A154118 A154119 A154120 * A154122 A154123 A154124

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 06 2009

STATUS

approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)