A226658 -id:A226658 - cp's OEIS frontend

A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 205, 65, 1, 1, 211, 1795, 1795, 211, 1, 1, 665, 14221, 36317, 14221, 665, 1, 1, 2059, 106819, 636331, 636331, 106819, 2059, 1, 1, 6305, 778765, 10365005, 23679901, 10365005, 778765, 6305, 1

Offset: 1

N. J. A. Sloane, Oct 04 2015

Two 1's are connected if they are in the same row or column. The requirement is for them to form a single connected set.
The number of m X n binary matrices with no zero rows or columns is given by A183109(m, n). If there are multiple components (not connected) then they cannot share either rows or columns. For i < n and j < m there are T(i,j) ways of creating an i X j component that occupies the first row. Its remaining i-1 rows may be on any of the remaining m-1 rows with its j columns on any of the n columns. The m-i rows and n-j columns not used by this component can be any matrix with no zero rows or columns.
T(m,n) is also the number of bipartite connected labeled graphs with parts of size m and n. (See A005333, A227322.)
This is the array a(m,n) in Kreweras (1969). Kreweras describes this as a symmetric triangle read by rows, giving numbers of connected relations.
The companion array b(m,n) (and the first few of its diagonals) in Kreweras (1969) should also be added to the OEIS if they are not already present.

			Table starts:
==========================================================================
m\n| 1    2      3         4           5             6               7
---|----------------------------------------------------------------------
1  | 1    1      1         1           1             1               1 ...
2  | 1    5     19        65         211           665            2059 ...
3  | 1   19    205      1795       14221        106819          778765 ...
4  | 1   65   1795     36317      636331      10365005       162470155 ...
5  | 1  211  14221    636331    23679901     805351531     26175881341 ...
6  | 1  665 106819  10365005   805351531   56294206205   3735873535339 ...
7  | 1 2059 778765 162470155 26175881341 3735873535339 502757743028605 ...
...
As a triangle, this begins:
  1;
  1,    1;
  1,    5,      1;
  1,   19,     19,      1;
  1,   65,    205,     65,      1;
  1,  211,   1795,   1795,    211,      1;
  1,  665,  14221,  36317,  14221,    665,    1;
  1, 2059, 106819, 636331, 636331, 106819, 2059, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..780
G. Kreweras, Inversion des polynômes de Bell bidimensionnels et application au dénombrement des relations binaires connexes, C. R. Acad. Sci. Paris Ser. A-B 268 1969 A577-A579.

Essentially same table as triangle A227322 (where the antidiagonals only go halfway).
Main diagonal is A005333.
Initial diagonals give A001047, A002501, A002502.
Cf. A226658, A123260, A286189, A183109, A048291, A287274.

A183109[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m-j) - 1)^n, {j, 0, m}];
T[m_, n_] := A183109[m, n] - Sum[T[i, j]*A183109[m - i, n - j] Binomial[m - 1, i - 1]*Binomial[n, j], {i, 1, m - 1}, {j, 1, n - 1}];
Table[T[m - n + 1, n], {m, 1, 9}, {n, 1, m}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

G(N)={my(S=matrix(N,N), T=matrix(N,N));
for(m=1,N,for(n=1,N,
S[m,n]=sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
T[m,n]=S[m,n]-sum(i=1, m-1, sum(j=1, n-1, T[i,j]*S[m-i,n-j]*binomial(m-1,i-1)*binomial(n,j)));
));T}
G(7) \\ Andrew Howroyd, May 22 2017

T(m,n) = A183109(m,n) - Sum_{i=1..m-1} Sum_{j=1..n-1} T(i,j)*A183109(m-i, n-j)*binomial(m-1,i-1)*binomial(n,j). - Andrew Howroyd, May 22 2017

Revised by N. J. A. Sloane, May 26 2017, to incorporate material from Andrew Howroyd's May 22 2017 submission (formerly A287297), which was essentially identical to this, although giving an alternative description and more information.

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.

Original entry on oeis.org

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

Pavel Irzhavski, Jul 06 2013

			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.

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

Cf. A226658, A123260.
Main diagonal gives: A005333.
Columns m=2, 3, 4 give: A001047, A002501, A002502.

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.)

cp's OEIS Frontend

A262307 Array read by antidiagonals: T(m,n) = number of m X n binary matrices with all 1's connected and no zero rows or columns.

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