A057546 -id:A057546 - cp's OEIS frontend

A082326 Permutation of natural numbers: A057163-conjugate of A057512.

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082325. a(n) = A069787(A082325(A069787(n))). a(n) = A082328(A082853(n))+A082852(n). Occurs in A073200 as row 1794. Cf. also A082337-A082338.
Differs from A082341 first time at n=39: a(39)=49, while A082341(39)=48.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

a(n) = A057163(A057512(A057163(n)))

A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 15, 36, 108, 301, 814, 2080, 5223, 12919, 32557, 83943, 222591, 600252, 1632814, 4440240, 12043224, 32572225, 88081208, 238722759, 649725756, 1776546687, 4877740703, 13432630929, 37063472432, 102389547753

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fifth row of A079216. The leftmost edge of the triangle A079221 and also its row sums shifted by one. Occurs in A073202 as row 9259542121261050623. Cf. A057546, A079223-A079227.

Maple
```
A079226 := n -> A079216bi(n,5);
```

a(n) = A079216(n, 5)

A082341 Permutation of natural numbers induced by the Catalan bijection gma082341 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This is A057163-conjugate of A073285.

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082342. a(n) = A057163(A073285(A057163(n))). Occurs in A073200 as row 1800. Cf. also A072797, A082337-A082339.
Differs from A082326 first time at n=39: a(39)=48, while A082326(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A243496 Matula-Goebel signature computed for the oriented trees that stay same when "deep-rotated": a(n) = A127301(A243495(n)), listed in the same order as those tree are encoded in A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 7, 5, 16, 9, 19, 17, 11, 32, 53, 23, 67, 59, 31, 64, 27, 49, 131, 25, 241, 83, 331, 277, 127, 128, 311, 103, 227, 739, 97, 1523, 431, 2221, 1787, 709, 256, 81, 361, 719, 169, 169, 289, 2063, 121, 563, 1433, 5623, 509, 12763, 3001, 19577, 15299, 5381, 512

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Note the first duplicate at a(43) = a(44) = 169, computed for two distinct fixed points in A243495: 1330 & 1535, which encode as A007088(A014486(1330)) = 1101100011100100 and A007088(A014486(1535)) = 1110010011011000 exactly those "dual cases" mentioned in the comments in A057546. Cf. also the comments at A243497.

Cf. A057546, A014486, A127301, A243495, A243497.

(define (A243496 n) (A127301 (A243495 n)))

A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 43, 104, 273, 702, 1870, 4985, 13562, 37038, 102266, 283774, 793189, 2227115, 6286044, 17811751, 50672898, 144639235, 414181050, 1189365940, 3424477813, 9883578364, 28589660227, 82870288432, 240672107114

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The third row of A079216. The leftmost edge of the triangle A079219 and also its row sums shifted by one. Occurs in A073202 as row 43639. Cf. A057546, A079223-A079227.

Maple
```
A079224 := n -> A079216bi(n,3);
```

a(n) = A079216(n, 3)

A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

Original entry on oeis.org

1, 1, 2, 5, 11, 30, 82, 233, 680, 2033, 6164, 18923, 58768, 184045, 581105, 1846906, 5905364, 18980465, 61292929, 198758704, 646974285, 2113163707, 6923642271, 22749608810, 74946337830, 247499313730, 819154110660, 2716779932308

Offset: 0

Antti Karttunen Jan 03 2002

nonn

The fourth row of A079216. The leftmost edge of the triangle A079220 and also its row sums shifted by one. Occurs in A073202 as row 2290625151. Cf. A057546, A079223-A079227.

Maple
```
A079225 := n -> A079216bi(n,4);
```

a(n) = A079216(n, 4)

A243497 A243496 sorted into ascending order, with duplicates removed.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 49, 53, 59, 64, 67, 81, 83, 97, 103, 121, 125, 127, 128, 131, 169, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 509, 512, 529, 563, 661, 691, 709, 719, 739, 841, 961, 1009, 1024, 1433, 1523, 1619, 1681, 1787, 1849, 1879, 2063

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Matula-codes for trees which are almost "uniform", but which allow cases like 169, 841, 1009, 1681, 1849, ... where there is a special relation between prime index and the exponent. (Cf. the comments at A243496).
Differs from A214577 for the first time at n=31, where A214577(31)=227, while here we have 169 at that position, because it corresponds exactly to that "dual" case mentioned in A057546, in excess to those mentioned in A003238. Note that 169 = 13*13 = p_{2*3}^2.
a(0) = 1 stands for the empty tree.

Cf. A214577 (a subsequence), A209638, A243494.

A082342 Permutation of natural numbers induced by the Catalan bijection gma082342 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 17 2003

nonn

This is A057163-conjugate of A073284.

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A082341. a(n) = A057163(A073284(A057163(n))). Occurs in A073200. Cf. also A072797, A082337-A082339.
Differs from A082325 first time at n=39: a(39)=48, while A082325(39)=49.
Number of cycles: A057513. Number of fixed-points: A057546. Max. cycle size: A000793. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A130968 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A130919/A130920.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 25, 54, 131, 331, 864, 2292, 6169, 16835

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

Cf. A130967, A057546.

cp's OEIS Frontend

A082326 Permutation of natural numbers: A057163-conjugate of A057512.

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A079226 Number of Catalan objects fixed by five-fold application of the Catalan bijections A057511/A057512 (deep rotation of general parenthesizations/plane trees).

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Maple

Formula

A082341 Permutation of natural numbers induced by the Catalan bijection gma082341 acting on the parenthesizations encoded by A014486/A063171.

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A243496 Matula-Goebel signature computed for the oriented trees that stay same when "deep-rotated": a(n) = A127301(A243495(n)), listed in the same order as those tree are encoded in A014486.

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Scheme

A079224 Number of Catalan objects fixed by three-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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Author

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Programs

Maple

Formula

A079225 Number of Catalan objects fixed by four-fold application of the Catalan bijections A057511/A057512 (Deep rotation of general parenthesizations/plane trees).

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Author

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Crossrefs

Programs

Maple

Formula

A243497 A243496 sorted into ascending order, with duplicates removed.

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Author

Keywords

Comments

Crossrefs

A082342 Permutation of natural numbers induced by the Catalan bijection gma082342 acting on the parenthesizations encoded by A014486/A063171.

Views

Author

Keywords

Comments

Links

Crossrefs

A130968 Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutations A130919/A130920.

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