A227322 Triangle read by rows: T(n, m) for 0 <= m <= n is the number of bipartite connected labeled graphs with parts of size n and m.

1, 1, 1, 0, 1, 5, 0, 1, 19, 205, 0, 1, 65, 1795, 36317, 0, 1, 211, 14221, 636331, 23679901, 0, 1, 665, 106819, 10365005, 805351531, 56294206205, 0, 1, 2059, 778765, 162470155, 26175881341, 3735873535339, 502757743028605

Offset: 0

Author

Pavel Irzhavski, Jul 06 2013

Keywords

• easy
• nonn
• tabl

Examples

			Triangle T(n, m) begins:
n\m 0 1    2      3         4           5             6               7
0   1
1   1 1
2   0 1    5
3   0 1   19    205
4   0 1   65   1795     36317
5   0 1  211  14221    636331    23679901
6   0 1  665 106819  10365005   805351531   56294206205
7   0 1 2059 778765 162470155 26175881341 3735873535339 502757743028605
...
Consider labeled bipartite graph with parts of size 2 and 2. To make graph connected it is possible to use all four possible edges or omit any one of them. Thus T(2, 2) = 5.
		

Links

• F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.

Crossrefs

Cf. A226658, A123260.

Main diagonal gives: A005333.

Columns m=2, 3, 4 give: A001047, A002501, A002502.

Formula

T(n, m) = 2^(n*m) - sum for all (i, j) in ({1, 2, ..., n} X {1, 2, ..., m} UNION (1, 0)) \ (n, m) \ (1 - n, 0) of T(i, j)*C(n - 1, i - 1)*C(m, j)*2^((n - i)*(m - j)), where C(n, m) is the binomial coefficient (A007318). This relation can be obtained considering connected component which contains the first vertex of the largest part. (If the largest part has zero size we get T(0, 0) = 2^0 - 0 = 1 which is true.)