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 1-9 of 9 results.

A089404 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A073286/A073287.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 74, 222, 698, 2264, 7536, 25600, 88406, 309440, 1095354, 3914340, 14102324, 51165004, 186774368, 685495334, 2527957006
Offset: 0

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Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Links

A073200 Number of simple Catalan bijections of type B.

Original entry on oeis.org

0, 1, 0, 3, 1, 0, 2, 2, 1, 0, 7, 3, 3, 1, 0, 8, 4, 2, 3, 1, 0, 6, 6, 8, 2, 3, 1, 0, 4, 5, 7, 7, 2, 3, 1, 0, 5, 7, 6, 6, 8, 2, 3, 1, 0, 17, 8, 5, 8, 7, 7, 2, 2, 1, 0, 18, 9, 4, 4, 6, 8, 7, 3, 3, 1, 0, 20, 10, 22, 5, 5, 5, 8, 4, 2, 2, 1, 0, 21, 14, 21, 17, 4, 4, 6, 5, 8, 3, 3, 1, 0
Offset: 0

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Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

Each row is a permutation of nonnegative integers induced by a Catalan bijection (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.
The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other Catalan bijection in this table, using one of the five allowed recursion types:
0 - Apply the given Catalan bijection and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of row number = 2)
1 - First recurse down to both subtrees of the old binary tree and only after that apply the given Catalan bijection. (last digit = 4)
2 - Apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)
3 - First recurse down to the right subtree of old binary tree and only after that apply the given Catalan bijection. (last digit = 8)
4 - First recurse down to the left subtree of old binary tree, after that apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree. (last digit = 0)
The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side and any Catalan bijection from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.

Links

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

Crossrefs

Four other tables giving the corresponding cycle-counts: A073201, counts of the fixed elements: A073202, the lengths of the largest cycles: A073203, the LCM's of all the cycles: A073204. The ordinary compositions are encoded using the N X N -> N bijection A054238 (which in turn uses the bit-interleaving function A000695).
The first 21 rows of this table:.
Row 0: A069770. Row 1: A072796. Row 2: A057163. Row 3: A073269, Row 4: A057163 (duplicate), Row 5: A073270, Row 6: A069767, Row 7: A001477 (identity perm.), Row 8: A069768, Row 9: A073280.
Row 10: A069770 (dupl.), Row 11: A072796 (dupl.), Row 12: A057511, Row 13: A073282, Row 14: A057512, Row 15: A073281, Row 16: A057509, Row 17: A073280 (dupl.), Row 18: A057510, Row 19: A073283, Row 20: A073284.
Other Catalan bijection-induced EIS-permutations which occur in this table. Only the first known occurrence is given. Involutions are marked with *, others paired with their inverse:.
Row 164: A057164*, Row 168: A057508*, Row 179: A072797*.
Row 41: A073286 - Row 69: A073287. Row 105: A073290 - Row 197: A073291. Row 416: A073288 - Row 696: A073289.
Row 261: A057501 - Row 521: A057502. Row 2618: A057503 - Row 5216: A057504. Row 2614: A057505 - Row 5212: A057506.
Row 10435: A073292 - Row ...: A073293. Row 17517: A057161 - Row ...: A057162.
For a more practical enumeration system of (some) Catalan automorphisms see table A089840 and its various "recursive derivations".

A069767 Signature-permutation of Catalan bijection "Knick".

Original entry on oeis.org

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

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Author

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

Keywords

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

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.

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

Examples

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Links

Crossrefs

Cf A127301, A153835, A153829.

Formula

a(n) = A127301(A057123(n)).
Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A073288 Permutation of natural numbers induced by the Catalan bijection gma073288! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073289. Occurs for first time in A073200 as row 416.
The scheme function gma073286! referred to below given in A073286.

A073287 Permutation of natural numbers induced by the composition of the Catalan bijections A069768 & A069770.

Original entry on oeis.org

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

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Author

Antti Karttunen, Jun 25 2002

Keywords

Links

Crossrefs

Inverse permutation: A073286. Occurs for first time in A073200 as row 69. Counts of the fixed elements: A073268.

Formula

a(n) = A069768(A069770(n)).

A073268 Number of plane binary trees whose right (or respectively: left) subtree is a unique "complete" tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.

Original entry on oeis.org

1, 1, 2, 3, 8, 20, 58, 179, 576, 1902, 6426, 22092, 77026, 271702, 967840, 3476555, 12578728, 45800278, 167693698, 617037126, 2280467586, 8461771342, 31510700712, 117725789124, 441141656810, 1657559677646, 6243810767912
Offset: 0

Views

Author

Antti Karttunen, Jun 25 2002

Keywords

Comments

The Catalan bijection A073286 fixes only these kinds of trees, so this occurs in A073202 as row 41.

Crossrefs

Occurs for first time in A073202 as row 41.

Programs

  • Maple
    A073268 := proc(n) local i; if(0=n) then 1 else add(Cat(n-2^i),i=0..floor(evalf(log[2](n)))); fi; end;
Cat := n -> binomial(2*n,n)/(n+1);
  • Mathematica
    a[0] = 1; a[n_] := Sum[CatalanNumber[n - 2^i], {i, 0, Log[2, n]}]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 05 2016 *)
  • PARI
    N=66; x='x+O('x^N); lg=ceil(log(N)/log(2));
C(x)=(1-sqrt(1-4*x))/(2*x);
gf=1+sum(k=0, lg, x^(2^k)*C(x) );
Vec(gf)
/* Joerg Arndt, Jul 02 2012 */

Formula

a(0)=1, a(n) = Sum_{i=0..floor(log_2(n))} Cat(n-(2^i))
G.f.: 1 + Sum_{k>=0} x^(2^k)*C(x) where C(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108). [Joerg Arndt, Jul 02 2012]

A130341 Row 3 of A122203.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 05 2007

Keywords

Comments

The signature-permutation of the Catalan automorphism which is derived from the third non-recursive Catalan automorphism *A089850 with recursion schema SPINE (see A122203 for the definition). This automorphism is also produced when automorphism *A069767 is applied to the right-hand side subtree of the given binary tree, with the left side left intact.

Links

Crossrefs

Inverse: A130342. a(n) = A069767(A069770(n)). The number of cycles and the number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A089404 and A073268. Cf. also A073286.

A122312 Row 7 of A122284.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2006

Keywords

Comments

The signature-permutation of the automorphism which is derived from the seventh non-recursive automorphism *A089854 with recursion schema NEPEED (see A122284 for the definition).

Links

Crossrefs

Inverse: A122311. Differs from A073286 for the first time at n=35, where a(n)=36, while A073286(n)=35.
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