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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

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.

FORMULA

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

EXAMPLE

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;

  ...

MATHEMATICA

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 *)

PROG

(PARI)

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

CROSSREFS

Essentially same table as triangle A227322 (where the antidiagonals only go half-way).

Main diagonal is A005333.

Initial diagonals give A001047, A002501, A002502.

Cf. A226658, A123260, A286189, A183109, A048291, A287274.

Sequence in context: A154334 A178346 A168551 * A144397 A047909 A171243

Adjacent sequences:  A262304 A262305 A262306 * A262308 A262309 A262310

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Oct 04 2015

EXTENSIONS

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.

STATUS

approved

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)