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.

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A130998 Signature permutation of a Catalan automorphism: row 19 of A122287.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 4, 5, 6, 20, 18, 22, 21, 17, 11, 10, 9, 12, 13, 14, 15, 19, 16, 57, 61, 50, 49, 55, 58, 63, 64, 62, 59, 46, 48, 45, 54, 30, 33, 27, 26, 29, 28, 23, 25, 34, 32, 24, 36, 35, 31, 39, 38, 37, 40, 41, 51, 52, 56, 60, 42, 43, 44, 47, 53, 183, 173, 192, 187, 169
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130358 with recursion scheme FORK.

Links

Crossrefs

Inverse: A130997.

A131000 Signature permutation of a Catalan automorphism: row 21 of A122287.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130360 with recursion scheme FORK.

Links

Crossrefs

Inverse: A130999.

A131002 Signature permutation of a Catalan automorphism: row 20 of A122287.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130362 with recursion scheme FORK.

Links

Crossrefs

Inverse: A131001.

A131004 Signature permutation of a Catalan automorphism: row 13 of A122287.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 5, 4, 17, 16, 18, 15, 14, 22, 19, 21, 12, 11, 20, 13, 10, 9, 45, 44, 46, 43, 42, 50, 47, 49, 40, 39, 48, 41, 38, 37, 62, 60, 63, 52, 51, 64, 56, 58, 31, 30, 57, 32, 29, 28, 61, 53, 59, 36, 33, 55, 35, 26, 25, 54, 34, 27, 24, 23, 129, 128, 130, 127, 126
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130364 with recursion scheme FORK.

Links

Crossrefs

Inverse: A131003.

A131006 Signature permutation of a Catalan automorphism: row 14 of A122287.

Original entry on oeis.org

0, 1, 3, 2, 6, 8, 7, 5, 4, 19, 15, 14, 20, 22, 16, 21, 18, 11, 13, 17, 12, 10, 9, 52, 53, 60, 39, 41, 51, 40, 38, 61, 55, 37, 54, 62, 64, 56, 43, 42, 57, 63, 47, 59, 48, 33, 29, 50, 28, 34, 36, 44, 58, 49, 30, 35, 46, 32, 25, 27, 45, 31, 26, 24, 23, 155, 151, 153, 179, 159
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130366 with recursion scheme FORK.

Links

Crossrefs

Inverse: A131005.

A131008 Signature permutation of a Catalan automorphism: row 16 of A122287.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 5, 4, 16, 14, 15, 18, 17, 19, 22, 20, 11, 12, 21, 13, 10, 9, 42, 44, 47, 38, 37, 43, 41, 39, 48, 49, 40, 50, 46, 45, 60, 51, 52, 62, 63, 53, 61, 57, 30, 28, 54, 29, 32, 31, 56, 64, 55, 33, 36, 59, 34, 25, 26, 58, 35, 27, 24, 23, 126, 121, 122, 131, 128
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A130368 with recursion scheme FORK.

Links

Crossrefs

Cf. A122287, A130368.
Inverse: A131007.

A131010 Signature permutation of a Catalan automorphism: row 17 of A122287.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 4, 6, 20, 17, 21, 22, 18, 13, 12, 10, 9, 11, 15, 14, 19, 16, 57, 61, 45, 48, 54, 58, 62, 63, 64, 59, 50, 49, 46, 55, 32, 36, 34, 31, 35, 26, 27, 24, 23, 25, 29, 28, 33, 30, 41, 40, 38, 37, 39, 52, 51, 56, 60, 43, 42, 47, 44, 53, 183, 169, 187, 192, 173
Offset: 0

Views

Author

Antti Karttunen, Jun 20 2007

Keywords

Comments

Derived from automorphism *A074686 with recursion scheme FORK.

Links

Crossrefs

Inverse: A131009.

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

Views

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.

A057164 Self-inverse permutation of natural numbers induced by reflections of the rooted plane trees and mountain ranges encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 11, 16, 19, 10, 15, 12, 17, 20, 13, 18, 21, 22, 23, 37, 28, 42, 51, 25, 39, 30, 44, 53, 33, 47, 56, 60, 24, 38, 29, 43, 52, 26, 40, 31, 45, 54, 34, 48, 57, 61, 27, 41, 32, 46, 55, 35, 49, 58, 62, 36, 50, 59, 63, 64, 65, 107, 79, 121, 149, 70
Offset: 0

Views

Author

Antti Karttunen, Aug 18 2000

Keywords

Comments

CatalanRankGlobal given in A057117 and the other Maple procedures in A056539.
Composition with A057163 gives Donaghey's Map M (A057505/A057506).

Examples

			a(10)=14 and a(14)=10, A014486[10] = 172 (10101100 in binary), A014486[14] = 202 (11001010 in binary) and these encode the following mountain ranges (and the corresponding rooted plane trees), which are reflections of each other:
...../\___________/\
/\/\/__\_________/__\/\/\
...
...../...........\
..\|/.............\|/

Links

Crossrefs

A057123(A057163(n)) = A057164(A057123(n)) for all n. Also the car/cdr-flipped conjugate of A069787, i.e., A057164(n) = A057163(A069787(A057163(n))). Fixed terms are given by A061856. Cf. also A057508, A069772.
Row 2 of tables A122287 and A122288.

Programs

  • Maple
    a(n) = CatalanRankGlobal(runcounts2binexp(reverse(binexp2runcounts(A014486[n])))) # i.e., reverse and complement the totally balanced binary sequences
  • PARI
    See Links section.

Formula

a(n) = A080300(A036044(A014486(n))) = A080300(A056539(A014486(n))).

A057506 Signature-permutation of a Catalan Automorphism: (inverse of) "Donaghey's map M", acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 11, 17, 12, 10, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 39, 54, 40, 38, 37, 63, 56, 57, 43, 42, 59, 47, 50, 36, 33, 48, 34, 29, 28, 58, 44, 49, 35, 30, 46, 32, 27, 25, 45, 31, 26, 24, 23, 196, 191, 192, 178, 177
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

This is inverse of A057505, which is a signature permutation of Catalan automorphism (bijection) known as "Donaghey's map M". See A057505 for more comments, links and references.

Links

Crossrefs

Inverse: A057505.
Cf. A057161, A057162, A057163, A057164, A057501, A057502, A057503, A057504 (for similar signature permutations of simple Catalan automorphisms).
Cf. A057507 (cycle counts).
The 2nd, 3rd, 4th, 5th and 6th "powers" of this permutation: A071662, A071664, A071666, A071668, A071670.
Row 12 of table A122287.
Cf. also A014486, A080300, A080069, A080070.

Programs

  • Maple
    map(CatalanRankGlobal,map(DonagheysA057506,CatalanSequences(196))); # Where CatalanSequences(n) gives the terms A014486(0..n).
DonagheysA057506 := n -> pars2binexp(deepreverse(DonagheysA057505(deepreverse(binexp2pars(n)))));
DonagheysA057505 := h -> `if`((0 = nops(h)), h, [op(DonagheysA057505(car(h))), DonagheysA057505(cdr(h))]);
# The following corresponds to automorphism A057164:
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
# The rest of required Maple-functions: see the given OEIS Wiki page.
  • Scheme
    (define (A057506 n) (CatalanRankSexp (*A057506 (CatalanUnrankSexp n))))
(define (*A057506 bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (*A057506 (car lt))))))))
;; Functions CatalanRankSexp and CatalanUnrankSexp can be found at OEIS Wiki page.

Formula

a(n) = A057163(A057164(n)).
a(n) = A057163(A057505(A057163(n))) = A057164(A057505(A057164(n))).

Extensions

Entry revised by Antti Karttunen, May 30 2017
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