A072797 Self-inverse permutation of natural numbers induced by a Catalan bijection acting on binary trees as encoded by A014486. See comments and examples for details.

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 11, 12, 13, 17, 18, 16, 14, 15, 20, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 39, 40, 41, 54, 55, 53, 51, 52, 57, 56, 58, 59, 61, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Author

Antti Karttunen, Jun 12 2002

Keywords

• nonn

Comments

This bijection 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 --> x B () A () A

\ / \ / \ / --> \ /

x x x x

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

See the example for an explanation of how to obtain a given integer sequence from this definition.

Notably for this permutation, A127301(a(n)) = A127301(n) does not always hold, even though for all n, A129593(a(n)) = A129593(n). - Antti Karttunen, Jan 14 2024

Examples

			To obtain the signature permutation, we apply these transformations to the binary trees as encoded and ordered by A014486 and for each n, a(n) will be the position of the tree to which the n-th tree is transformed to, as follows:
.
                   one tree of one internal
  empty tree         (non-leaf) node
      x                      \/
n=    0                      1
a(n)= 0                      1               (both are always fixed)
.
the next 7 trees, with 2-3 internal nodes, in range [A014137(1), A014137(2+1)-1] = [2,8] are:
.
                          \/     \/                 \/     \/
       \/     \/         \/       \/     \/ \/     \/       \/
      \/       \/       \/       \/       \_/       \/       \/
n=     2        3        4        5        6        7        8
.
and the new shapes after swapping the two subtrees in positions marked "B" and "C" in the diagram given in the comments are:
.
                          \/     \/       \/               \/
       \/     \/         \/       \/     \/       \/ \/     \/
      \/       \/       \/       \/       \/       \_/       \/
a(n)=  2        3        4        5        7        6        8
thus we obtain the first nine terms of this sequence: 0, 1, 2, 3, 4, 5, 7, 6, 8.
		

Links

• A. Karttunen, Catalan Automorphisms
• Index entries for signature-permutations induced by Catalan automorphisms

Crossrefs

Row 8 of A089840.

Cf. A014486, A057163, A072796, A127301, A129593.

Counts for the fixed points and for the number of distinct cycles (in each subrange limited by A014137) are given by A073190 and A073191.

Formula

a(n) = A057163(A072796(A057163(n))).

Extensions

Further comments added by Antti Karttunen, Jun 04 2011 and Mar 30 2024