A000840 -id:A000840 - cp's OEIS frontend

A333159 Triangle read by rows: T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column up to permutation of rows and columns.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 5, 4, 1, 1, 1, 1, 4, 12, 12, 4, 1, 1, 1, 1, 7, 31, 66, 31, 7, 1, 1, 1, 1, 8, 90, 433, 433, 90, 8, 1, 1, 1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1, 1, 1, 14, 938, 30404, 171984, 171984, 30404, 938, 14, 1, 1

Offset: 0

Andrew Howroyd, Mar 10 2020

Rows and columns may be permuted independently. The case that rows and columns must be permuted together is covered by A333161.

T(n,k) is the number of k-regular bicolored graphs on 2n unlabeled nodes which are invariant when the two color classes are exchanged.

			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,   5,    4,    1,    1;
  1, 1,  4,  12,   12,    4,    1,   1;
  1, 1,  7,  31,   66,   31,    7,   1,  1;
  1, 1,  8,  90,  433,  433,   90,   8,  1, 1;
  1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1;
  ...
The T(2,1) = 1 matrix is:
  [1 0]
  [0 1]
.
The T(4,2)= 2 matrices are:
  [1 1 0 0]   [1 1 0 0]
  [1 1 0 0]   [1 0 1 0]
  [0 0 1 1]   [0 1 0 1]
  [0 0 1 1]   [0 0 1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..230

Columns k=0..4 are A000012, A000012, A002865, A000840, A000843.
Row sums are A333160.
Central coefficients are A333165.
Cf. A008327, A122082, A333157, A133687, A333161.

T(n,k) = T(n,n-k).

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.

Original entry on oeis.org

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

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

A000843 Number of quartic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 12, 66, 433, 3442, 30404, 294951, 3093770, 34846185, 419020738, 5355289843, 72466668921, 1034818990771, 15548633703424, 245182764121059, 4047990858838185, 69826392559499108, 1256005988628521464, 23517396850332614602, 457623646061902648705

Offset: 1

Brendan McKay and Eric Rogoyski

nonn

Column k=4 of A333159.

Cf. A000513, A000840.

a(11)-a(25) from Andrew Howroyd, Mar 10 2020

cp's OEIS Frontend

A333159 Triangle read by rows: T(n,k) is the number of non-isomorphic n X n symmetric binary matrices with k ones in every row and column up to permutation of rows and columns.

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

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A000843 Number of quartic bicolored graphs on n unlabeled nodes admitting an automorphism exchanging the colors.

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