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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A181153 Number of connected 7-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 8, 741, 2887493
Offset: 0

Views

Author

Jason Kimberley, last week of Jan 2011

Keywords

Comments

a(10) was computed by the author in 3 hours using GENREG on Dec 02 2009.
a(11) was computed by the author using GENREG over 45.7 processor days at U. Newcastle from Jan 25 to 27 2011.

Examples

			The a(0)=1 null graph is vacuously 7-regular and connected; since it is acyclic then it has infinite girth.
The a(7)=1 graph is the complete bipartite graph K_{7,7} on 14 vertices.
The a(8)=1 graph has girth 4, automorphism group of order 80640, and the following adjacency lists:
01 : 02 03 04 05 06 07 08
02 : 01 09 10 11 12 13 14
03 : 01 09 10 11 12 13 15
04 : 01 09 10 11 12 14 15
05 : 01 09 10 11 13 14 15
06 : 01 09 10 12 13 14 15
07 : 01 09 11 12 13 14 15
08 : 01 10 11 12 13 14 15
09 : 02 03 04 05 06 07 16
10 : 02 03 04 05 06 08 16
11 : 02 03 04 05 07 08 16
12 : 02 03 04 06 07 08 16
13 : 02 03 05 06 07 08 16
14 : 02 04 05 06 07 08 16
15 : 03 04 05 06 07 08 16
16 : 09 10 11 12 13 14 15

References

  • M. Meringer, Fast Generation of Regular Graphs and Construction of Cages. Journal of Graph Theory, 30 (1999), 137-146.

Links

Crossrefs

7-regular simple graphs with girth at least 4: this sequence (connected), A185274 (disconnected), A185374 (not necessarily connected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), this sequence (k=7), A181154 (k=8), A181170 (k=9).
Connected 7-regular simple graphs with girth at least g: A014377 (g=3), this sequence (g=4).
Connected 7-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4).

A181170 Number of connected 9-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 14
Offset: 0

Views

Author

Jason Kimberley, last week of Jan 2011

Keywords

Comments

a(11)=14 was computed by the author using GENREG at U. Ncle. over 615 processor days during Dec 2009.

Examples

			The a(0)=1 null graph is vacuously 8-regular and connected; since it is acyclic then it has infinite girth.
The a(9)=1 graph is the complete bipartite graph K_{9,9} with 18 vertices.
The a(10)=1 graph has girth 4, automorphism group of order 7257600, and the following adjacency lists:
01 : 02 03 04 05 06 07 08 09 10
02 : 01 11 12 13 14 15 16 17 18
03 : 01 11 12 13 14 15 16 17 19
04 : 01 11 12 13 14 15 16 18 19
05 : 01 11 12 13 14 15 17 18 19
06 : 01 11 12 13 14 16 17 18 19
07 : 01 11 12 13 15 16 17 18 19
08 : 01 11 12 14 15 16 17 18 19
09 : 01 11 13 14 15 16 17 18 19
10 : 01 12 13 14 15 16 17 18 19
11 : 02 03 04 05 06 07 08 09 20
12 : 02 03 04 05 06 07 08 10 20
13 : 02 03 04 05 06 07 09 10 20
14 : 02 03 04 05 06 08 09 10 20
15 : 02 03 04 05 07 08 09 10 20
16 : 02 03 04 06 07 08 09 10 20
17 : 02 03 05 06 07 08 09 10 20
18 : 02 04 05 06 07 08 09 10 20
19 : 03 04 05 06 07 08 09 10 20
20 : 11 12 13 14 15 16 17 18 19

Links

Crossrefs

9-regular simple graphs with girth at least 4: this sequence (connected), A185294 (disconnected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), A058275 (k=5), A058276 (k=6), A181153 (k=7), A181154 (k=8), this sequence (k=9).
Connected 9-regular simple graphs with girth at least g: A014378 (g=3), this sequence (g=4).
Connected 9-regular simple graphs with girth exactly g: A184993 (g=3).

A157989 The numbers of the jingle for a popular Ontario pizza chain's phone number.

Original entry on oeis.org

9, 6, 7, 11, 11
Offset: 1

Views

Author

Dave [Last name?] (dave(AT)endicott.net), Mar 10 2009

Keywords

Comments

Sung as: "nine, six, seven, eleven, eleven, .... ".
Jingle first introduced in 1978 and has since become a pop culture item for Canadians.

Links

Crossrefs

Cf. A104175.

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.

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