A069767 Signature-permutation of Catalan bijection "Knick".

0, 1, 3, 2, 7, 8, 6, 5, 4, 17, 18, 20, 21, 22, 16, 19, 15, 12, 13, 14, 11, 10, 9, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 38, 29, 26, 27, 37, 28, 25, 24, 23, 129, 130, 132, 133, 134

Offset

0,3

Comments

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153141. See further comments there.

This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.

Note: the name in Finnish is "Niks".

References

A. Karttunen, paper in preparation.

Links

A. Karttunen, Table of n, a(n) for n = 0..2055

Index entries for signature-permutations induced by Catalan bijections

Prog

(Scheme implementations of this automorphism. These act on S-expressions, i.e. list-structures:)
(CONSTRUCTIVE VERSION:) (define (*A069767 s) (cond ((not (pair? s)) s) (else (cons (cdr s) (*A069767 (car s))))))
(DESTRUCTIVE VERSION:) (define (*A069767! s) (cond ((pair? s) (*A069770! s) (*A069767! (cdr s)))) s)

Crossrefs

Inverse permutation: "Knack", A069768. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073290, A073292, A073294, A073296, A073298.

In range [A014137(n-1)..A014138(n-1)] of this permutation, the number of cycles is A073431, number of fixed points: A036987 (Fixed points themselves: A084108), Max. cycle size & LCM of all cycle sizes: A011782. See also: A074080.

A127302(a(n)) = A127302(n) for all n. a(n) = A057508(A057161(n)) = A057161(A069769(n)).

Row 1 of A122203 and A122287, row 15 of A122286 and A130403, row 6 of A073200.

See also bijections A073286, A082345, A082348, A082349, A130341.

Sequence in context: A122351 A323710 A089863 * A154458 A127378 A122302

Adjacent sequences: A069764 A069765 A069766 * A069768 A069769 A069770

Keyword

nonn

Author

Antti Karttunen, Apr 16 2002; entry revised Dec 20 2008

Status

approved