A154121 Signature permutation of a Catalan bijection: row 3655 of A089840.

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

Offset: 0

Author

Antti Karttunen, Jan 06 2009

Keywords

• nonn

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

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
• J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson's groups, L'Enseignement Mathématique, Vol. 42 (1996), pp. 215-256.
• A. Karttunen, On the connection of A089840 with Thompson's groups, speculation and conjectures.
• Index entries for signature-permutations of Catalan automorphisms

Crossrefs

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