A000843 -id:A000843

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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.

+10
12

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

Triangle begins:

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]

CROSSREFS

Columns k=0..4 are A000012, A000012, A002865, A000840, A000843.

Row sums are A333160.

Central coefficients are A333165.

Cf. A008327, A122082, A333157, A133687, A333161.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Mar 10 2020

STATUS

approved

A000840

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..30.

CROSSREFS

Column k=3 of A333159.

Cf. A000512, A000843, A004066, A008325.

KEYWORD

nonn

AUTHOR

Brendan McKay and Eric Rogoyski

EXTENSIONS

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

STATUS

approved

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