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 1-7 of 7 results.

A153831 Index sequence to A089840: set-wise difference of A153829 and A153830.

Original entry on oeis.org

68, 73, 74, 83, 84, 87, 88, 183, 184, 189, 190, 199, 202, 203, 252, 254, 261, 262, 268, 269, 270, 271, 515, 537, 539, 573, 575, 591, 593, 871, 894, 895, 990, 995, 996, 1110, 1132, 1134, 1466, 1489, 1490, 1585, 1590, 1591, 1600, 1601, 1604, 1605, 2213
Offset: 0

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Author

Antti Karttunen, Jan 07 2009

Keywords

Comments

The terms give the positions to bijections in A089840 which preserve A153835/A127302 (the non-oriented form of binary trees), but do not extend uniquely to automorphisms of an infinite binary tree.

Links

Crossrefs

Cf. also A153826, A153827, A153828, A153832, A153833.

A089840 Signature permutations of non-recursive Catalan automorphisms (i.e., bijections of finite plane binary trees, with no unlimited recursion down to indefinite distances from the root), sorted according to the minimum number of opening nodes needed in their defining clauses.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 10, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 14, 13, 12, 8, 7, 6
Offset: 0

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Author

Antti Karttunen, Dec 05 2003; last revised Jan 06 2009

Keywords

Comments

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

References

  • A. Karttunen, paper in preparation, draft available by e-mail.

Links

Crossrefs

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A072796, 3: A089850, 4: A089851, 5: A089852, 6: A089853, 7: A089854, 8: A072797, 9: A089855, 10: A089856, 11: A089857, 12: A074679, 13: A089858, 14: A073269, 15: A089859, 16: A089860, 17: A074680, 18: A089861, 19: A073270, 20: A089862, 21: A089863.
Other rows: row 83: A154125, row 169: A129611, row 183: A154126, row 251: A129612, row 253: A123503, row 258: A123499, row 264: A123500, row 3608: A129607, row 3613: A129605, row 3617: A129606, row 3655: A154121, row 3656: A154123,row 3702: A082354, row 3747: A154122, row 3748: A154124, row 3886: A082353, row 4069: A082351, row 4207: A089865, row 4253: A082352, row 4299: A089866, row 65167: A129609, row 65352: A129610, row 65518: A123495, row 65796: A123496, row 79361: A123492, row 1653002: A123695, row 1653063: A123696, row 1654023: A073281, row 1654249: A123498, row 1654694: A089864, row 1654720: A129604,row 1655089: A123497, row 1783367: A123713, row 1786785: A123714.
Tables A122200, A122201, A122202, A122203, A122204, A122283, A122284, A122285, A122286, A122287, A122288, A122289, A122290, A130400-A130403 give various "recursive derivations" of these non-recursive automorphisms. See also A089831, A073200.
Index sequences to this table, giving various subgroups or other important constructions: A153826, A153827, A153829, A153830, A123694, A153834, A153832, A153833.

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

A153826 Index sequence to A089840: positions of bijections that preserve A127301 (the non-oriented form of general trees).

Original entry on oeis.org

0, 2, 22, 23, 24, 25, 26, 91, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 395, 531, 634, 876, 1005, 1109, 1228, 1229, 1230, 1231, 1232, 1704, 3608, 3611, 3613, 3615, 3617, 4392
Offset: 0

Views

Author

Antti Karttunen, Jan 07 2009

Keywords

Comments

These terms form a subgroup in A089840 (A089839). Because A127301 can be computed as a fold and most of the recursive derivations of A089840 (i.e., tables A122201-A122204, A122283-A122290, A130400-A130403) are also folds, this sequence also gives the indices to those derived tables where bijections preserving A127301 occur.

Links

Crossrefs

Subset of A153827. Apart from 0, has no other terms in common with A153829. Cf. also A153828, A153830, A153831, A153832, A153833.

A153830 Index sequence to A089840: positions of bijections that preserve A127302 (the non-oriented form of binary trees) and whose behavior does not depend on whether there are internal or terminal nodes (leaves) in the neighborhood of any vertex.

Original entry on oeis.org

0, 1, 3, 7, 15, 21, 27, 46, 92, 114, 149, 169, 225, 251, 299, 400, 638, 753, 1233, 1348, 1705, 1823, 1992, 2097, 2335, 2451, 2995, 3128, 3485, 3607, 3677, 3771, 4214, 4307, 4631, 5254, 6692, 7393, 10287, 10988, 13145, 13860, 20353, 21054
Offset: 0

Views

Author

Antti Karttunen, Jan 07 2009

Keywords

Comments

These elements form a subgroup in A089840 (A089839) isomorphic to a group consisting of all finitely iterated wreath products of the form S_2 wr S_2 wr ... wr S_2 and each is an image of some finitary automorphism of an infinite binary tree. E.g. A089840(1) = *A069770 is an image of the generator A of Grigorchuk Group. See comments at A153246 and A153141.
The defining properties are propagated by all recursive transformations of A089840 which themselves do not behave differently depending whether there are internal or terminal vertices in the neighborhood of any vertex (at least the ones given in A122201-A122204, A122283-A122290, A130400-A130403), so this sequence gives also the corresponding positions in those tables.

Links

Crossrefs

Subset of A153829. Cf. also A153831, A153826, A153827, A153828, A153832, A153833.

A153827 Index sequence to A089840: positions of bijections that preserve A129593 (that is, they permute the Łukasiewicz-word computed for a general tree).

Original entry on oeis.org

0, 2, 8, 22, 23, 24, 25, 26, 45, 71, 91, 115, 119, 121, 125, 127, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 395, 396, 397, 398, 399, 514, 525, 526, 531, 532, 633, 634, 635, 636, 637
Offset: 0

Views

Author

Antti Karttunen, Jan 07 2009

Keywords

Comments

These elements form a subgroup in A089840 (A089839).

Links

Crossrefs

Superset of A153826.
Cf. A153828, A153829, A153830, A153831, A153832, A153833.

A153828 Index sequence to A089840: set-wise difference of A153827 and A153826.

Original entry on oeis.org

8, 45, 71, 115, 119, 121, 125, 127, 396, 397, 398, 399, 514, 525, 526, 532, 633, 635, 636, 637, 656, 657, 658, 659, 660, 661, 752, 757, 758, 874, 880, 888, 892, 993, 1001, 1120, 1121, 1126, 1127, 1156, 1157, 1168, 1169, 1174, 1175, 1347, 1394, 1395
Offset: 0

Views

Author

Antti Karttunen, Jan 07 2009

Keywords

Comments

The terms give the positions of bijections in A089840 which preserve A129593, but not A127301.

Links

Crossrefs

Cf. also A153829, A153830, A153831, A153832, A153833.
Showing 1-7 of 7 results.

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