A008300 -id:A008300 - cp's OEIS frontend

A067209 Number of n X n 0..1 matrices with all row and column sums equal.

1, 2, 4, 14, 140, 4322, 434542, 144109562, 165431317452, 654143160457922, 9331115832294448754, 469229841129645179004962, 87481268294773501231007850158, 58328998963405322376273800187396962, 147013017541698957281403494604070439260442

Offset: 0

R. H. Hardin, Feb 20 2002

nonn

			Some solutions for n=5
..1..1..0..0..1....0..0..1..0..1....0..1..0..1..1....1..1..0..1..0
..1..1..0..0..1....1..1..0..0..0....1..1..0..0..1....1..0..1..1..0
..0..0..1..1..1....1..0..0..1..0....1..1..1..0..0....1..0..1..0..1
..1..0..1..1..0....0..1..1..0..0....0..0..1..1..1....0..1..0..1..1
..0..1..1..1..0....0..0..0..1..1....1..0..1..1..0....0..1..1..0..1

Alois P. Heinz, Table of n, a(n) for n = 0..30
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums
B. D. McKay, 0-1 matrices with constant row and column sums

Row sums of A008300.

Column k=1 of A202784.

Added one term and example - R. H. Hardin, Jan 05 2012

a(0)=1 prepended and more terms (using data provided by B. D. McKay) from Alois P. Heinz, Apr 12 2017

A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.

Original entry on oeis.org

1, 2, 90, 297200, 116963796250, 6736218287430460752, 64051375889927380035549804336, 108738182111446498614705217754614976371200, 34812290428176298285394893936773707951192224124239796250, 2188263032066768922535710968724036448759525154977348944382853301460850000

Offset: 0

David desJardins, Dec 22 2000

nonn

Vladeta Jovovic, Nov 12 2006, Table of n, a(n) for n = 0..15
A Conflitti, C. M. Da Fonseca, and R. Mamede, The maximal length of a chain in the Bruhat order for a chain of binary matrices., Lin. Algebra Applic. (2011)
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the largest size of an antichain in the Bruhat order for A(2k, k).
Alessandro Conflitti, C. M. da Fonseca and Ricardo Mamede, On the Largest Size of an Antichain in the Bruhat Order for A(2k,k), ORDER, 2011, DOI: 10.1007/s11083-011-9241-1.
Jonathan Jedwab and Tabriz Popatia, A new representation of mutually orthogonal frequency squares, Simon Fraser University (Burnaby, BC, Canada, 2020).
M. A. Khojastepour and M. Farajzadeh-Tehrani, Characterizing per Node Degrees of Freedom in an Interference Network, 2014; also 2014 IEEE International Symposium on Information Theory, pp. 1016-1020.
B. D. McKay, 0-1 matrices with constant row and column sums
Michael Penn, A not so magic square..., YouTube video, 2021.
Wikipedia, Dynamic programming

Central coefficients of A008300.
Main diagonal of A376935.
Cf. A001499, A001501, A253316.

More terms (using dynamic programming in Python) from Greg Kuperberg, Feb 08 2001

More terms from Vladeta Jovovic, Nov 12 2006

A075754 Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.

Original entry on oeis.org

1, 720, 3110940, 24046189440, 315031400802720, 6736218287430460752, 226885231700215713535680, 11649337108041078980732943360, 885282776210120715086715619724160, 96986285294151066094112970262797953280

Offset: 5

Michel Buffet (buffet(AT)engref.fr), Oct 08 2002

nonn

Also number of ways to arrange 5n rooks on an n X n chessboard, with no more than 5 rooks in each row and column. - Vaclav Kotesovec, Aug 04 2013

Generally (Canfield + McKay, 2004), a(n) ~ exp(-1/2)*binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi)*exp(-n*s-1/2*(s-1)^2)*(n*s)^(n*s+1/2)*(s!)^(-2*n). - Vaclav Kotesovec, Aug 04 2013

B. D. McKay, Applications of a technique for labeled enumeration, Congressus Numerantium, 40 (1983) 207-221.

Vaclav Kotesovec, Table of n, a(n) for n = 5..61, (computed with program by Doron Zeilberger, see link below)
B. D. McKay, 0-1 matrices with constant row and column sums
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums.
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses, Maple package Bipartite.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for sequences related to binary matrices

Column 5 of A008300.

From Vaclav Kotesovec, Aug 04 2013: (Start)
a(n) ~ exp(-1/2)*binomial(n,5)^(2*n) / binomial(n^2,5*n), (Canfield + McKay, 2004)
a(n) ~ sqrt(Pi)*2^(1/2-6*n)*5^(3*n+1/2) *9^(-n)*exp(-5*n-8)*n^(5*n+1/2)
(End)

More terms from Brendan McKay, Jan 08 2005

A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

Original entry on oeis.org

2, 7, 16, 34, 265, 512, 209, 7343, 41503, 65536, 1546, 304186, 6474726, 24997921, 33554432, 13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736, 130922, 1336221251, 702998475376, 34215495252681, 252540841305558, 505874809287625

Offset: 1

R. J. Mathar, Mar 09 2017

			Triangle begins:
2;
7,     16;
34,    265,      512;
209,   7343,     41503,      65536;
1546,  304186,   6474726,    24997921,    33554432;
13327, 17525812, 1709852332, 21252557377, 57366997447, 68719476736;
...

Andrew Howroyd, Table of n, a(n) for n = 1..84

Cf. A002720 (column k=1), A197458 (column k=2), A008300 (exactly k 1s).
Main diagonal and first lower diagonal give: A002416, A048291.
Cf. A247158 (k=n/2).

A172541 Number of n X n 0..1 arrays with row sums 7 and column sums 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 40320, 14398171200, 8302816499443200, 7722015017013984456000, 11649337108041078980732943360, 28278447454165011203551734584421120, 108738182111446498614705217754614976371200

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..23

Column 7 of A008300, Cf. A075754.

A172544 Number of n X n 0..1 arrays with row sums 6 and column sums 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5040, 187530840, 12025780892160, 1289144584143523800, 226885231700215713535680, 64051375889927380035549804336, 28278447454165011203551734584421120

Offset: 1

R. H. Hardin Feb 06 2010

nonn

Also number of ways to arrange 6n rooks on an n X n chessboard, with no more than 6 rooks in each row and column. - Vaclav Kotesovec, Aug 04 2013

Generally (Canfield + McKay, 2004), a(n) ~ exp(-1/2)*binomial(n,s)^(2*n) / binomial(n^2,s*n), or a(n) ~ sqrt(2*Pi)*exp(-n*s-1/2*(s-1)^2)*(n*s)^(n*s+1/2)*(s!)^(-2*n). - Vaclav Kotesovec, Aug 04 2013

R. H. Hardin, Table of n, a(n) for n=1..28
E. R. Canfield and B. D. McKay, Asymptotic enumeration of dense 0-1 matrices with equal row and column sums, Electron. J. Combin. 12 (2005)
Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can n (Straight) Men and n (Straight) Women Get Married, if Each Person Has Exactly k Spouses

Column 6 of A008300.

From Vaclav Kotesovec, Aug 04 2013: (Start)
a(n) ~ exp(-1/2)*binomial(n,6)^(2*n)/binomial(n^2,6*n), (Canfield + McKay, 2004)
a(n) ~ sqrt(Pi)*2^(1-2*n)*3^(2*n+1/2)*5^(-2*n)*exp(-6*n-25/2)*n^(6*n+1/2)
(End)

A172534 Number of n X n 0..1 arrays with row sums 12 and column sums 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6227020800, 676508133623135814000, 65662040698002721810659005184000, 6892692735539278753058456514221737762215000

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..18

Column k=12 of A008300.

A172535 Number of n X n 0..1 arrays with row sums 11 and column sums 11.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 479001600, 3574340599104475200, 27537152449960680597739468800, 254143667822686635850590661555095468000, 3183529624645847695375078143769686741065620316160

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..18

Column k=11 of A008300.

A172536 Number of n X n 0..1 arrays with row sums 8 and column sums 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 362880, 1371785398200, 7673688777463632000, 65599839591251908982712750, 885282776210120715086715619724160, 19040419266278799766631032461849139013040

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..22

Column k=8 of A008300.

A172537 Number of n X n 0..1 arrays with row sums 14 and column sums 14.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1307674368000, 36574751938491748341360000, 665825560532772251175492202972938240000, 10431401634793817906193873163767479249710797568200000

Offset: 1

R. H. Hardin, Feb 06 2010

nonn

R. H. Hardin, Table of n, a(n) for n=1..19

Column k=14 of A008300.

cp's OEIS Frontend

A067209 Number of n X n 0..1 matrices with all row and column sums equal.

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A058527 Number of 2n X 2n 0-1 matrices with n ones in each row and each column.

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A075754 Number of n X n (0,1) matrices containing exactly five 1's in each row and in each column.

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A283500 Triangle read by rows: T(n,k) = number of n X n (0,1) matrices with at most k 1's in each row or column.

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A172541 Number of n X n 0..1 arrays with row sums 7 and column sums 7.

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A172544 Number of n X n 0..1 arrays with row sums 6 and column sums 6.

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A172534 Number of n X n 0..1 arrays with row sums 12 and column sums 12.

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A172535 Number of n X n 0..1 arrays with row sums 11 and column sums 11.

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A172536 Number of n X n 0..1 arrays with row sums 8 and column sums 8.

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A172537 Number of n X n 0..1 arrays with row sums 14 and column sums 14.

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