A073299 -id:A073299 - cp's OEIS frontend

A069768 Signature-permutation of Catalan bijection "Knack".

0, 1, 3, 2, 8, 7, 6, 4, 5, 22, 21, 20, 17, 18, 19, 16, 14, 9, 10, 15, 11, 12, 13, 64, 63, 62, 58, 59, 61, 57, 54, 45, 46, 55, 48, 49, 50, 60, 56, 53, 44, 47, 51, 42, 37, 23, 24, 38, 25, 26, 27, 52, 43, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 196, 195, 194, 189, 190

Offset: 0

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

nonn

This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left 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 A153142. See further comments there and in A153141.
This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(*A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) 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 row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack.
Note: the name in Finnish is "Naks".

A. Karttunen, paper in preparation.

Inverse permutation: "Knick", A069767. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073291, A073293, A073295, A073297, A073299.
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) = A057162(A057508(n)) = A069769(A057162(n))
Row 1 of A122204 and A122288, row 21 of A122285 and A130402, row 8 of A073200.
See also bijections A073287, A082346, A082347, A082350, A130342.

A073290 Permutation A069767 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 12, 13, 11, 10, 9, 15, 14, 19, 21, 22, 16, 20, 18, 17, 31, 32, 34, 35, 36, 30, 33, 29, 26, 27, 28, 25, 24, 23, 40, 41, 39, 38, 37, 52, 51, 56, 58, 59, 60, 62, 63, 64, 43, 42, 53, 57, 61, 47, 55, 49, 50, 44, 54, 48, 46, 45, 87, 88, 90, 91, 92, 96, 97

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073291. Occurs for first time in A073200 as row 105. A073267 gives essentially (apart from the first two terms) the counts of elements fixed. Cf. A073292-A073299.

a(n) = A069767(A069767(n)).

A073292 Permutation A069767 applied three times ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 21, 22, 20, 18, 17, 19, 16, 14, 10, 9, 15, 11, 13, 12, 58, 59, 62, 63, 64, 57, 61, 55, 49, 50, 54, 48, 46, 45, 56, 60, 53, 47, 44, 51, 42, 38, 26, 27, 37, 25, 24, 23, 52, 43, 39, 29, 28, 41, 33, 35, 36, 40, 30, 34, 32, 31, 170, 171, 174, 175, 176

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073293. Occurs for first time in A073200 as row 10435. Cf. A073290-A073299.

a(n) = A069767(A069767(A069767(n))).

A073291 Permutation A069768 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 13, 12, 11, 9, 10, 15, 14, 19, 22, 21, 16, 20, 17, 18, 36, 35, 34, 31, 32, 33, 30, 28, 23, 24, 29, 25, 26, 27, 41, 40, 39, 37, 38, 52, 51, 60, 64, 63, 56, 62, 58, 59, 43, 42, 53, 61, 57, 44, 54, 45, 46, 47, 55, 48, 49, 50, 106, 105, 104, 100, 101

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073290. Occurs for first time in A073200 as row 197. Cf. A073292-A073299.

a(n) = A069768(A069768(n)).

A073293 Permutation A069768 applied three times ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 18, 17, 20, 22, 21, 16, 19, 15, 13, 12, 14, 11, 9, 10, 50, 49, 48, 45, 46, 55, 54, 61, 64, 63, 57, 62, 58, 59, 47, 44, 53, 60, 56, 43, 52, 41, 36, 35, 40, 34, 31, 32, 42, 51, 39, 33, 30, 37, 28, 23, 24, 38, 29, 25, 26, 27, 148, 147, 146, 142, 143

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073292. Cf. A073290-A073299.

a(n) = A069768(A069768(A069768(n))).

A073297 Permutation A069768 applied five times or composition of the permutations A073291 & A073293.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 6, 4, 5, 21, 22, 20, 18, 17, 19, 16, 14, 10, 9, 15, 11, 13, 12, 59, 58, 62, 64, 63, 57, 61, 55, 50, 49, 54, 48, 45, 46, 56, 60, 53, 47, 44, 51, 42, 38, 27, 26, 37, 25, 23, 24, 52, 43, 39, 29, 28, 41, 33, 36, 35, 40, 30, 34, 31, 32, 176, 175, 174, 170, 171

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073296. Cf. A073290-A073299.

a(n) = A069768^5(n) = A073291(A073293(n)).

A073294 Permutation A069767 applied four times or permutation A073290 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 13, 12, 14, 15, 16, 18, 17, 19, 20, 22, 21, 26, 27, 25, 24, 23, 29, 28, 33, 35, 36, 30, 34, 32, 31, 38, 37, 39, 41, 40, 42, 43, 47, 49, 50, 44, 48, 46, 45, 51, 52, 53, 55, 54, 60, 61, 63, 64, 56, 57, 62, 59, 58, 73, 74, 76, 77, 78, 72, 75

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073295. Cf. A073290-A073299. Occurs in A073200 as row 142647371.

a(n) = A069767(A069767(A069767(A069767(n)))) = A073290(A073290(n)).

A073295 Permutation A069768 applied four times or permutation A073291 applied twice.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 11, 13, 12, 14, 15, 16, 18, 17, 19, 20, 22, 21, 27, 26, 25, 23, 24, 29, 28, 33, 36, 35, 30, 34, 31, 32, 38, 37, 39, 41, 40, 42, 43, 47, 50, 49, 44, 48, 45, 46, 51, 52, 53, 55, 54, 60, 61, 64, 63, 56, 57, 62, 58, 59, 78, 77, 76, 73, 74, 75, 72

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073294. Cf. A073290-A073299.

a(n) = A069768(A069768(A069768(A069768(n)))) = A073291(A073291(n)).

A073296 Permutation A069767 applied five times or composition of the permutations A073290 & A073292.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 6, 5, 4, 18, 17, 20, 22, 21, 16, 19, 15, 13, 12, 14, 11, 9, 10, 49, 50, 48, 46, 45, 55, 54, 61, 63, 64, 57, 62, 59, 58, 47, 44, 53, 60, 56, 43, 52, 41, 35, 36, 40, 34, 32, 31, 42, 51, 39, 33, 30, 37, 28, 24, 23, 38, 29, 25, 27, 26, 142, 143, 146, 147, 148

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073297. Cf. A073290-A073299. Occurs in A073200 as row 36169536789029067.

a(n) = A069767^5(n) = A073290(A073292(n)).

A073298 Permutation A069767 applied six times or permutation A073290 "cubed" or permutation A073292 "squared".

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 13, 12, 11, 9, 10, 15, 14, 19, 22, 21, 16, 20, 17, 18, 35, 36, 34, 32, 31, 33, 30, 28, 24, 23, 29, 25, 27, 26, 41, 40, 39, 37, 38, 52, 51, 60, 63, 64, 56, 62, 59, 58, 43, 42, 53, 61, 57, 44, 54, 46, 45, 47, 55, 48, 50, 49, 100, 101, 104, 105, 106

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A073299. Cf. A073290-A073297. Occurs in A073200 as row 2596188043348680205834878944583883.

a(n) = A069767^6(n) = A073290(A073290(A073290(n))) = A073292(A073292(n))

cp's OEIS Frontend

A069768 Signature-permutation of Catalan bijection "Knack".

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A073290 Permutation A069767 applied twice ("squared").

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A073292 Permutation A069767 applied three times ("cubed").

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A073291 Permutation A069768 applied twice ("squared").

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A073293 Permutation A069768 applied three times ("cubed").

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A073297 Permutation A069768 applied five times or composition of the permutations A073291 & A073293.

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A073294 Permutation A069767 applied four times or permutation A073290 applied twice.

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A073295 Permutation A069768 applied four times or permutation A073291 applied twice.

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A073296 Permutation A069767 applied five times or composition of the permutations A073290 & A073292.

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A073298 Permutation A069767 applied six times or permutation A073290 "cubed" or permutation A073292 "squared".

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