cp's OEIS Frontend

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.......B...A

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

...A...x...-->... .x...C...............A..().........()..A..

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

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

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

See the Karttunen OEIS-Wiki link for a detailed explanation how to obtain a given integer sequence from this definition.

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).

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

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

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

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

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

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

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123503 with the recursion schema SPINE (defined in A122203).

The number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation begins as 1,1,2,1,3,1,4,1,8,1,16,1,47,..., the LCM of cycle sizes as 1,1,1,2,12,12,120,120,840,840,5040,5040,55440,... (cf. A089423) and the cycle-count sequence seems to be A045629. (To be proved.)