A376935 -id:A376935 - cp's OEIS frontend

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1

Offset: 0

Andrew Howroyd, May 09 2020

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.

			Array begins:
====================================================================
n\k | 0 1   2     3       4          5            6            7
----|---------------------------------------------------------------
  0 | 1 1   1     1       1          1            1            1 ...
  1 | 1 1   1     1       1          1            1            1 ...
  2 | 1 1   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]

Andrew Howroyd, Table of n, a(n) for n = 0..350 (first 26 antidiagonals)

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
Main diagonal is A172645.
Cf. A008300, A333901, A376935, A377063 (up to row permutations).

T(n,k) = T(k,n).

A377007 Array read by antidiagonals: T(n,k) is the number of inequivalent 2n X 2k binary matrices with all row sums k and column sums n up to permutations of rows and columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 4, 7, 4, 1, 1, 1, 1, 5, 19, 19, 5, 1, 1, 1, 1, 7, 46, 194, 46, 7, 1, 1, 1, 1, 8, 132, 3144, 3144, 132, 8, 1, 1, 1, 1, 10, 345, 65548, 601055, 65548, 345, 10, 1, 1, 1, 1, 12, 951, 1272696, 128665248, 128665248, 1272696, 951, 12, 1, 1

Offset: 0

Andrew Howroyd, Oct 12 2024

Terms may be computed without generating each matrix by enumerating the number of matrices by column sum sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A376935. Burnside's lemma can be used to extend this method to the unlabeled case. This seems to require looping over partitions for both rows and columns.

			Array begins:
============================================================================
n\k | 0 1 2   3       4           5               6                    7 ...
----+-----------------------------------------------------------------------
  0 | 1 1 1   1       1           1               1                    1 ...
  1 | 1 1 1   1       1           1               1                    1 ...
  2 | 1 1 2   3       4           5               7                    8 ...
  3 | 1 1 3   7      19          46             132                  345 ...
  4 | 1 1 4  19     194        3144           65548              1272696 ...
  5 | 1 1 5  46    3144      601055       128665248          24124134235 ...
  6 | 1 1 7 132   65548   128665248    294494683312      607662931576945 ...
  7 | 1 1 8 345 1272696 24124134235 607662931576945 14584161564179926207 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..119 (first 15 antidiagonals)

Main diagonal is A333740.
Columns k=0..6 are A000012, A000012, A001399, A377003, A377004, A377005, A377006.
Cf. A133687, A376935.

T(n,k) = T(k,n).

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

A172555 Number of 2*n X 8 binary arrays with row sums 4 and column sums n.

Original entry on oeis.org

1, 70, 44730, 60871300, 116963796250, 273957842462220, 729833528645228700, 2128370421607383353400, 6637298299496411164868250, 21796558191507153941744357500, 74580431332898028185458459628980, 263863554962023852690547420699517000, 959794291357783500093926796260896462300

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..150 (terms n=1..33 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=4 of A376935.

a(0)=1 prepended by Christoph Koutschan, Oct 12 2024

A172556 Number of 2*n X 6 binary arrays with row sums 3 and column sums n.

Original entry on oeis.org

1, 20, 1860, 297200, 60871300, 14367744720, 3718394156400, 1026608414145600, 297348703692826500, 89365729945562642000, 27658131940039664137360, 8766913970029589057611200, 2834492178580299130305958000, 931874436031756882451601080000, 310768686646948895430510472680000

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..387 (terms n=1..49 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=3 of A376935.

(n+3) * (n+4)^5 * (33*n^2 + 176*n + 236) * a(n+4) = 2 * (n+3) * (2*n + 7) * (3201*n^6 + 61886*n^5 + 497179*n^4 + 2124170*n^3 + 5089654*n^2 + 6484024*n + 3431096) * a(n+3) + 16 * (2*n + 5) * (2*n + 7) * (2772*n^6 + 48048*n^5 + 344379*n^4 + 1307394*n^3 + 2775099*n^2 + 3125336*n + 1460132) * a(n+2) - 128 * (n+2) * (2*n + 3) * (2*n + 5) * (2*n + 7) * (7491*n^4 + 84898*n^3 + 351364*n^2 + 628997*n + 414370) * a(n+1) + 51200 * (n+1) * (n+2) * (2*n + 1) * (2*n + 3) * (2*n + 5) * (2*n + 7) * (33*n^2 + 242*n + 445) * a(n). - Doron Zeilberger and Christoph Koutschan, Oct 13 2024

a(0)=1 prepended by Andrew Howroyd, Oct 12 2024

A172557 Number of 2*n X 10 binary arrays with row sums 5 and column sums n.

Original entry on oeis.org

1, 252, 1172556, 14367744720, 273957842462220, 6736218287430460752, 195428754450309450171024, 6368685647203261761403856832, 226226183763370772622682749527820, 8588121524476528848476120449733261520, 343706681395653993021794459883981791479056, 14359560914153299239163671050863513324047517632

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..50 (terms n=1..24 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=5 of A376935.

a(0)=1 prepended by Christoph Koutschan, Oct 12 2024

A172558 Number of 2*n X 12 binary arrays with row sums 6 and column sums n.

Original entry on oeis.org

1, 924, 32496156, 3718394156400, 729833528645228700, 195428754450309450171024, 64051375889927380035549804336, 24197157773366540262083203214418624, 10156485647319165206190655366818708639900, 4623755659802671553551698062433487687144266000, 2245303346871705713053772485416749032101579706759056

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..39 (terms n=1..19 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=6 of A376935.

a(0)=1 prepended by Christoph Koutschan, Oct 12 2024

A172554 Number of 2*n X 18 binary arrays with row sums 9 and column sums n.

Original entry on oeis.org

1, 48620, 842090474940, 89365729945562642000, 21796558191507153941744357500, 8588121524476528848476120449733261520, 4623755659802671553551698062433487687144266000, 3100343219257976442736156588732505699555692275498897600, 2446263257880905042021634694206746192857701260816933527219717500

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..22 (terms n=1..12 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=9 of A376935.

a(0)=1 prepended by Andrew Howroyd, Oct 12 2024

A172559 Number of 2*n X 14 binary arrays with row sums 7 and column sums n.

Original entry on oeis.org

1, 3432, 936369720, 1026608414145600, 2128370421607383353400, 6368685647203261761403856832, 24197157773366540262083203214418624, 108738182111446498614705217754614976371200, 553419789083994907139302477834526824481001555000

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..30 (terms n=1..16 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=7 of A376935.

a(0)=1 prepended by Andrew Howroyd, Oct 12 2024

A172560 Number of 2*n X 16 binary arrays with row sums 8 and column sums n.

Original entry on oeis.org

1, 12870, 27770358330, 297348703692826500, 6637298299496411164868250, 226226183763370772622682749527820, 10156485647319165206190655366818708639900, 553419789083994907139302477834526824481001555000, 34812290428176298285394893936773707951192224124239796250

Offset: 0

R. H. Hardin, Feb 06 2010

nonn

Christoph Koutschan, Table of n, a(n) for n = 0..25 (terms n=1..13 from R. H. Hardin)
Robert Dougherty-Bliss, Christoph Koutschan, Natalya Ter-Saakov, and Doron Zeilberger, The (Symbolic and Numeric) Computational Challenges of Counting 0-1 Balanced Matrices, arXiv:2410.07435 [math.CO], 2024.

Column k=8 of A376935.

a(0)=1 prepended by Andrew Howroyd, Oct 12 2024

cp's OEIS Frontend

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

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A377007 Array read by antidiagonals: T(n,k) is the number of inequivalent 2*n X 2*k binary matrices with all row sums k and column sums n up to permutations of rows and columns.

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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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A172555 Number of 2*n X 8 binary arrays with row sums 4 and column sums n.

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A172556 Number of 2*n X 6 binary arrays with row sums 3 and column sums n.

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A172557 Number of 2*n X 10 binary arrays with row sums 5 and column sums n.

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A172558 Number of 2*n X 12 binary arrays with row sums 6 and column sums n.

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A172554 Number of 2*n X 18 binary arrays with row sums 9 and column sums n.

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A172559 Number of 2*n X 14 binary arrays with row sums 7 and column sums n.

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A172560 Number of 2*n X 16 binary arrays with row sums 8 and column sums n.

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A377007 Array read by antidiagonals: T(n,k) is the number of inequivalent 2n X 2k binary matrices with all row sums k and column sums n up to permutations of rows and columns.