A008325 -id:A008325 - cp's OEIS frontend

A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

0, 0, 1, 1, 2, 7, 16, 51, 224, 1165, 7454, 56349, 481309, 4548786, 46829325, 519812910, 6177695783, 78190425826, 1049510787100, 14886252250208, 222442888670708, 3492326723315796, 57468395960854710, 989052970923320185, 17767732298980160822, 332572885090541084172, 6475438355244504235759, 130954580036269713385884

Offset: 1

Eric Rogoyski

Also, isomorphism classes of bicolored cubic bipartite graphs, where isomorphism cannot exchange the colors.

			n=4: every matrix with 3 1's in each row and column can be transformed by permutation of rows (or columns) into {1110,1101,1011,0111}, therefore a(4)=1. - _Michael Steyer_, Feb 20 2003

A. Burgess, P. Danziger, E. Mendelsohn, B. Stevens, Orthogonally Resolvable Cycle Decompositions, 2013; http://www.math.ryerson.ca/~andrea.burgess/OCD-submit.pdf
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.

Index entries for sequences related to Latin squares and rectangles

Column k=3 of A133687.
Cf. A000186, A000513, A000840, A001501, A008325, A232215.
A079815 may be an erroneous version of this, or it may have a slightly different (as yet unknown) definition. - N. J. A. Sloane, Sep 04 2010.

Definition corrected by Brendan McKay, May 28 2006
a(1)-a(12) checked by Brendan McKay, Aug 27 2010
Terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

A006823 Number of connected trivalent bipartite graphs with 2n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 38, 149, 703, 4132, 29579, 245627, 2291589, 23466857, 259974248, 3087698618, 39075020582, 524492748500, 7439833931266, 111179110620163, 1745593888011626, 28726140345832290, 494406533261250407, 8881990236308687928, 166255686198172837111, 3237191621055524796848, 65467841616499410561220

Offset: 3

N. J. A. Sloane and Brendan McKay

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Brinkmann, Fast generation of cubic graphs, Journal of Graph Theory, 23(2):139-149, 1996.
Nicolas Gastineau and Olivier Togni, S-Packing Colorings of Cubic Graphs, arXiv:1403:7495 [cs.DM], 2016.
Jan Goedgebeur and Patric R. J. Ostergard, Switching 3-Edge-Colorings of Cubic Graphs, arXiv:2105:01363 [math.CO], May 2021. See Table 2.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
House of Graphs, Cubic bipartite graphs
M. Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Bicubic Graph [From _Eric W. Weisstein_, Jul 28 2009]

Column 3 of A008326.

Cf. A006824, A008325 (Euler transform).

Terms a(17) and beyond from Andrew Howroyd, Apr 03 2020

A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 6, 4, 1, 1, 1, 1, 4, 14, 14, 4, 1, 1, 1, 1, 7, 41, 130, 41, 7, 1, 1, 1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1, 1, 1, 12, 725, 62616, 304496, 62616, 725, 12, 1, 1, 1, 1, 14, 4196, 2806508, 78322916

Offset: 0

Brendan McKay

This sequence can be derived from A008326 by Euler transform. - Andrew Howroyd, Apr 03 2020

			Triangle begins:
  1,
  1, 1,
  1, 1, 1,
  1, 1, 1,   1,
  1, 1, 2,   1,    1,
  1, 1, 2,   2,    1,    1,
  1, 1, 4,   6,    4,    1,   1;
  1, 1, 4,  14,   14,    4,   1, 1;
  1, 1, 7,  41,  130,   41,   7, 1, 1;
  1, 1, 8, 157, 1981, 1981, 157, 8, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..189
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.

Column k=0..5 are A000012, A000012, A002865, A008325, A333730, A333731.
Row sums are A008324.
Cf. A051031, A008326, A087114, A133687, A333159.

Column k is the Euler transform of column k of A008326. - Andrew Howroyd, Apr 03 2020

More terms from Eric Rogoyski, May 15 1997

Name clarified by Andrew Howroyd, Sep 05 2018

A000840 Number of cubic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 12, 31, 90, 285, 938, 3285, 11983, 45390, 177803, 718390, 2986407, 12749364, 55802982, 250068732, 1145923828, 5363795830, 25620207380, 124767647097, 618983876918, 3126035142910, 16060182735947, 83883575376862, 445164927249466, 2399098651337048

Offset: 1

Brendan McKay and Eric Rogoyski

nonn

Column k=3 of A333159.

Cf. A000512, A000843, A004066, A008325.

a(11)-a(30) from Andrew Howroyd, Mar 10 2020

A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

Original entry on oeis.org

0, 0, 1, 1, 2, 6, 14, 41, 157, 725, 4196, 29817, 246646, 2297088, 23503564, 260265650, 3090341095, 39101587595, 524783295041, 7443251159470, 111222017297268, 1746166043555813, 28734210790531045, 494526547845483641, 8883866458982018870, 166286444108288113541, 3237719185652343485853, 65477290060076644381373

Offset: 1

Gunnar Brinkmann, Brendan McKay and Eric Rogoyski

nonn

Sean A. Irvine, On the difference between A004066 and A008325
Patric R. J. Östergård, Classifying generalized Howell designs, Designs Codes Cryptog. (2025). See Sect. 3.2.1.

Cf. A000512, A000840, A008325 (bipartite), A006823 (connected).

a(n) = (A000840(n) + A000512(n))/2. - Andrew Howroyd, Apr 01 2020

a(1)-a(2) prepended and terms a(15) and beyond from Andrew Howroyd, Apr 01 2020

cp's OEIS Frontend

A000512 Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

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A006823 Number of connected trivalent bipartite graphs with 2n nodes.

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A008327 Triangle read by rows: T(n,k) is the number of simple regular bipartite graphs with 2n nodes and degree k, (0 <= k <= n).

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A000840 Number of cubic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

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A004066 Number of simple regular trivalent bicolored graphs with 2n nodes.

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