cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A073201 Array of cycle count sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 4, 1, 1, 1, 22, 11, 3, 1, 1, 1, 66, 31, 7, 2, 1, 1, 1, 217, 96, 22, 4, 3, 1, 1, 1, 715, 305, 66, 11, 7, 2, 1, 1, 1, 2438, 1007, 217, 30, 22, 4, 2, 2, 1, 1, 8398, 3389, 715, 93, 66, 11, 3, 5, 1, 1, 1, 29414, 11636, 2438, 292, 217, 30, 6, 14, 2, 2, 1, 1
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row of this table gives the counts of separate orbits/cycles to which the Catalan bijection given in the corresponding row of A073200 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
Note that for involutions (self-inverse Catalan bijections) this is always (A000108(n)+Affffff(n))/2, where Affffff is the corresponding "fix-count sequence" from the table A073202.

Links

  • A. Karttunen, Gatomorphisms (With the complete source and explanation)

Crossrefs

Only the first known occurrence(s) given (marked with ? if not yet proved/unclear): rows 0, 2, 4, etc.: A007595, Row 1: A073191, Rows 6 (& 8): A073431, Row 7: A000108, Rows 12, 14, 20, ...: A057513, Rows 16, 18, ...: A003239, Row 57, ..., 164: A007123, Row 168: A073193, Row 261: A002995, Row 2614: A057507, Row 2618 (?), row 17517: A001683.

A089858 Permutation of natural numbers induced by Catalan Automorphism *A089858 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 4, 5, 14, 15, 19, 21, 22, 16, 20, 17, 9, 10, 18, 11, 12, 13, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 23, 24, 46, 25, 26, 27, 47, 55, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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.

Links

Crossrefs

Row 13 of A089840. Inverse of A089861. a(n) = A072797(A069770(n)) = A069770(A089852(n)) = A057163(A073270(A057163(n))).
Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089858) added by Antti Karttunen, Jun 04 2011

A089861 Permutation of natural numbers induced by Catalan Automorphism *A089861 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 6, 5, 17, 18, 20, 21, 22, 9, 10, 14, 16, 19, 11, 15, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 32, 33, 41, 34, 35, 36, 129, 130, 132, 133, 134, 138
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

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.

Links

Crossrefs

Row 18 of A089840. Inverse of A089858. a(n) = A089852(A069770(n)) = A069770(A072797(n)) = A057163(A073269(A057163(n))).
Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A082349 Permutation of natural numbers induced by the Catalan bijection gma082349 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 5, 4, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 22, 11, 10, 9, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 31, 32, 59, 34, 35, 36, 60, 61, 62, 30, 33, 63, 29, 26, 27, 64, 28, 25, 24, 23, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees left, if possible, otherwise applies Catalan bijection A069767.

Links

Crossrefs

Inverse of A082350. Cf. also A074679-A074680, A082335-A082336.
Number of cycles: A073193 (to be checked). Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082350 Permutation of natural numbers induced by the Catalan bijection gma082350 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Comments

This Catalan bijection rotates binary trees right, if possible, otherwise applies Catalan bijection A069768.

Links

Crossrefs

Inverse of A082349. Cf. also A074679-A074680, A082335-A082336.
Number of cycles: A073193 (to be checked). Number of fixed-points: A019590. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A073192 Number of general plane trees whose n-th subtree from the left is equal to the n-th subtree from the right, for all its subtrees (i.e., are palindromic in the shallow sense).

Original entry on oeis.org

1, 1, 2, 3, 8, 18, 54, 155, 500, 1614, 5456, 18630, 64960, 228740, 814914, 2926323, 10589916, 38561814, 141219432, 519711666, 1921142832, 7129756188, 26555149404, 99228108222, 371886574632, 1397548389644, 5265131346368
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

The Catalan bijection A057508 fixes only these kinds of trees, so this occurs in the table A073202 as row 168.

Links

Crossrefs

Occurs for first time in A073202 as row 168.
Cf. also A073190.

Programs

  • Maple
    A073192 := proc(n) local d; add( (`mod`((n-d+1),2))*Cat((n-d)/2)*(`if`((0=d),1,Cat(d-1))), d=0..n); end;
Cat := n -> binomial(2*n,n)/(n+1);
  • Mathematica
    a[n_] := Sum[Mod[n - k + 1, 2]*CatalanNumber[(n - k)/2]*If[k == 0, 1, CatalanNumber[k - 1]], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2016 *)
  • PARI
    Gat(n) = if (n == -1, 1, binomial(2*n,n)/(n+1));
a(n) = sum(i=0, n, if (!((n-i)%2), Gat((n-i)/2)*Gat(i-1))); \\ Michel Marcus, May 30 2018

Formula

a(n) = Sum_{i=0..n, (n-i) is even} Gat((n-i)/2)*Gat(i-1), where Gat(-1) = 1 and otherwise like A000108(n).
A073193(n) = (A000108(n) + A073192(n))/2.

A130373 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the first element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 10, 16, 19, 14, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 30, 33, 37, 29, 26, 44, 47, 27, 53, 56, 60, 28, 39, 38, 43, 52, 42, 40, 31, 45, 46, 32, 48, 49, 50, 51, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 66, 72, 75, 79, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 5 ... 2n 2n+1), permutes it as (1 3 2 5 4 ... 2n+1 2n).

Links

Crossrefs

Cf. A057508, A057164, A057164.
SPINE and ENIPS transform of *A130340 (transformations explained in A122203 and A122204).
The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.

Formula

a(n) = A057508(A130374(A057508(n))) = A057164(A130374(A057164(n))).

A130374 Signature permutation of a Catalan automorphism: flip the positions of even- and odd-indexed elements at the top level of the list, leaving the last element in place if the length (A057515(n)) is odd.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 14, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 24, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 66, 70, 72, 75, 67, 71
Offset: 0

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

This self-inverse automorphism permutes the top level of a list of even length (1 2 3 4 ... 2n-1 2n) as (2 1 4 3 ... 2n 2n-1), and when applied to a list of odd length (1 2 3 4 ... 2n-1 2n 2n+1), permutes it as (2 1 4 3 ... 2n 2n-1 2n+1).

Links

Crossrefs

Cf. a(n) = A057508(A130373(A057508(n))) = A057164(A130373(A057164(n))) = A127285(A127288(n)) = A127287(A127286(n)). Also a(A085223(n)) = A130370(A122282(A130369(A085223(n)))) holds for all n>=0. The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A073193 and A073192.
Showing 1-8 of 8 results.

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