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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

Is there an equivalent Catalan bijection in A073200?

Is there an equivalent Catalan bijection in A073200?

Is there an equivalent Catalan bijection in A073200?

Is there an equivalent Catalan bijection in A073200?

This automorphism either swaps (if A057515(n) > 1) the first two toplevel elements (of a general plane tree, like *A072796 does) and otherwise (if n > 1, A057515(n)=1) swaps the sides of the left hand side subtree of the S-expression (when viewed as a binary tree, like *A089854 does). This is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.

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

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

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

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

...x....(A072796)....x....................x...(A089854)...x

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

This is the first multiclause automorphism in table A089840 which cannot be represented as a composition of two smaller nonrecursive automorphisms, the property which is also shared by *A123499 and *A123500.

This self-inverse automorphism is obtained as either SPINE(*A129607) or ENIPS(*A129607). See the definitions given in A122203 and A122204.

Is there an equivalent Catalan bijection in A073200?

Is there an equivalent Catalan bijection in A073200?

Is there an equivalent Catalan bijection in A073200?