A333159 - OEIS

%I #5 Mar 11 2020 18:06:52

%S 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,

%T 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,

%U 7937,3442,285,12,1,1,1,1,14,938,30404,171984,171984,30404,938,14,1,1

%N 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.

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

%C 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.

%H Andrew Howroyd, <a href="/A333159/b333159.txt">Table of n, a(n) for n = 0..230</a>

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

%e Triangle begins:

%e   1;

%e   1, 1;

%e   1, 1,  1;

%e   1, 1,  1,   1;

%e   1, 1,  2,   1,    1;

%e   1, 1,  2,   2,    1,    1;

%e   1, 1,  4,   5,    4,    1,    1;

%e   1, 1,  4,  12,   12,    4,    1,   1;

%e   1, 1,  7,  31,   66,   31,    7,   1,  1;

%e   1, 1,  8,  90,  433,  433,   90,   8,  1, 1;

%e   1, 1, 12, 285, 3442, 7937, 3442, 285, 12, 1, 1;

%e   ...

%e The T(2,1) = 1 matrix is:

%e   [1 0]

%e   [0 1]

%e .

%e The T(4,2)= 2 matrices are:

%e   [1 1 0 0]   [1 1 0 0]

%e   [1 1 0 0]   [1 0 1 0]

%e   [0 0 1 1]   [0 1 0 1]

%e   [0 0 1 1]   [0 0 1 1]

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

%Y Row sums are A333160.

%Y Central coefficients are A333165.

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

%K nonn,tabl

%O 0,13

%A _Andrew Howroyd_, Mar 10 2020