cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A135588 Number of symmetric (0,1)-matrices with exactly n entries equal to 1 and no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 6, 20, 74, 302, 1314, 6122, 29982, 154718, 831986, 4667070, 27118610, 163264862, 1013640242, 6488705638, 42687497378, 288492113950, 1998190669298, 14177192483742, 102856494496050, 762657487965086, 5771613810502002, 44555989658479726, 350503696871063138
Offset: 0

Views

Author

Vladeta Jovovic, Feb 25 2008, Mar 03 2008, Mar 04 2008

Keywords

Examples

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(4) = 20 matrices:
  [11]
  [11]
.
  [110][101][100][100][011][010][010][001][001]
  [100][010][011][001][100][110][101][010][001]
  [001][100][010][011][100][001][010][101][110]
.
  [1000][1000][1000][1000][0100][0100][0010][0010][0001][0001]
  [0100][0100][0010][0001][1000][1000][0100][0001][0100][0010]
  [0010][0001][0100][0010][0010][0001][1000][1000][0010][0100]
  [0001][0010][0001][0100][0001][0010][0001][0100][1000][1000]
(End)

Links

Crossrefs

Row sums of A135589.
Cf. A049311, A054976, A101370, A104601, A104602, A120733, A138178, A283877, A316983, A320796, A321401, A321405.

Programs

  • Mathematica
    Table[Sum[SeriesCoefficient[(1+x)^k*(1+x^2)^(k*(k-1)/2)/2^(k+1),{x,0,n}],{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Jul 02 2014 *)
Join[{1},  Table[Length[Select[Subsets[Tuples[Range[n], 2], {n}], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Sort[Reverse/@#]==#]&]], {n, 5}]] (* Gus Wiseman, Nov 14 2018 *)

Formula

G.f.: Sum_{n>=0} (1+x)^n*(1+x^2)^binomial(n,2)/2^(n+1).
G.f.: Sum_{n>=0} (Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*(1+x)^k*(1+x^2)^binomial(k,2)).

A321720 Number of non-normal (0,1) semi-magic squares with sum of entries equal to n.

Original entry on oeis.org

1, 1, 2, 6, 25, 120, 726, 5040, 40410, 362881, 3630840, 39916800, 479069574, 6227020800, 87181402140, 1307674370040, 20922977418841, 355687428096000, 6402388104196400, 121645100408832000, 2432903379962038320, 51090942171778378800, 1124000886592995642000, 25852016738884976640000
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Comments

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Links

Crossrefs

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.
Cf. A321717, A321719, A321722, A321723.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Formula

a(p) = p! for p prime as the squares are all permutation matrices of order p and a(n) >= n! for n > 1 (see comments in A321717 and A321719). - Chai Wah Wu, Jan 13 2019
a(n) = Sum_{d|n, d<=n/d} A008300(n/d, d) for n > 0. - Andrew Howroyd, Apr 11 2020

Extensions

a(7) from Chai Wah Wu, Jan 13 2019
a(8)-a(15) from Chai Wah Wu, Jan 14 2019
a(16)-a(21) from Chai Wah Wu, Jan 16 2019
Terms a(22) and beyond from Andrew Howroyd, Apr 11 2020

A321723 Number of non-normal magic squares whose entries are all 0 or 1 and sum to n.

Original entry on oeis.org

1, 1, 0, 0, 9, 20, 96, 656, 5584, 48913, 494264, 5383552, 65103875, 840566080, 11834159652, 176621049784, 2838040416201, 48060623405312
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Comments

A non-normal magic square is a square matrix with row sums, column sums, and both diagonals all equal to d, for some d|n.

Examples

			The a(4) = 9 magic squares:
  [1 1]
  [1 1]
.
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0][0 1 0 0]
  [0 0 0 1][0 1 0 0][1 0 0 0][0 0 1 0][0 1 0 0][0 0 0 1][0 0 1 0][1 0 0 0]
  [0 1 0 0][0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0][0 0 1 0]

Links

Crossrefs

Cf. A006052, A007016, A057151, A068313, A101370, A104602, A120732, A271103, A319056, A319616.
Cf. A321717, A321718, A321719, A321720, A321721, A321722.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],SameQ@@Join[{Tr[prs2mat[#]],Tr[Reverse[prs2mat[#]]]},Total/@prs2mat[#],Total/@Transpose[prs2mat[#]]]]&]],{n,5}]

Formula

a(n) >= A007016(n) with equality if n is prime. - Chai Wah Wu, Jan 15 2019

Extensions

a(7)-a(15) from Chai Wah Wu, Jan 15 2019
a(16)-a(17) from Chai Wah Wu, Jan 16 2019

A321735 Number of (0,1)-matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 2, 7, 30, 153, 939, 6653, 53743, 486576
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Examples

			The a(3) = 7 matrices:
  [1 1]
  [1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Crossrefs

Cf. A000700, A007016, A049311, A054976, A057151, A104602, A320451, A321719, A321723, A321732, A321733, A321736, A321739.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Formula

Let c(y) be the coefficient of m(y) in e(y), where m is monomial symmetric functions and e is elementary symmetric functions. Then a(n) = Sum_{|y| = n} c(y).

A321739 Number of non-isomorphic weight-n set multipartitions (multisets of sets) whose part-sizes are also their vertex-degrees.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 21, 46, 94, 208
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

Also the number of (0,1) square matrices up to row and column permutations with n ones and no zero rows or columns, with the same multiset of row sums as of column sums.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 12 set multipartitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {1}{23}{23}      {12}{13}{23}
               {1}{2}{3}  {1}{1}{23}    {2}{13}{23}      {3}{23}{123}
                          {1}{3}{23}    {3}{3}{123}      {1}{1}{1}{234}
                          {1}{2}{3}{4}  {1}{2}{2}{34}    {1}{1}{24}{34}
                                        {1}{2}{4}{34}    {1}{2}{34}{34}
                                        {1}{2}{3}{4}{5}  {1}{3}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {3}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}

Crossrefs

Cf. A000700, A049311, A057151, A104602, A319056, A320451, A321719, A321721, A321723, A321732, A321734, A321735, A321736, A321854.

A321733 Number of (0,1)-matrices with n ones, no zero rows or columns, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 2, 8, 40, 246, 1816, 15630, 153592, 1696760, 20816358, 280807868, 4131117440, 65823490088, 1129256780408
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Examples

			The a(4) = 40 matrices:
  [1 1]
  [1 1]
.
  [1 1 0][1 1 0][1 0 1][1 0 1][1 0 0]
  [1 0 0][0 0 1][1 0 0][0 1 0][0 1 1]
  [0 0 1][1 0 0][0 1 0][1 0 0][0 1 0]
.
  [1 0 0][0 1 1][0 1 0][0 1 0][0 1 0]
  [0 0 1][1 0 0][1 1 0][1 0 1][0 1 1]
  [0 1 1][1 0 0][0 0 1][0 1 0][1 0 0]
.
  [0 1 0][0 0 1][0 0 1][0 0 1][0 0 1]
  [0 0 1][1 1 0][1 0 0][0 1 0][0 0 1]
  [1 0 1][0 1 0][0 1 1][1 0 1][1 1 0]
.
  [1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0]
  [0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][0 1 0 0][0 0 0 1][0 1 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][0 1 0 0][0 0 1 0][0 1 0 0]
.
  [0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0]
  [1 0 0 0][1 0 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 0 1 0][1 0 0 0]
.
  [0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 0 1][0 0 0 1]
  [0 1 0 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0]
  [0 0 0 1][0 1 0 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0]
.
  [0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0]
  [0 1 0 0][0 0 1 0][1 0 0 0][0 0 1 0][1 0 0 0][0 1 0 0]
  [0 0 1 0][0 1 0 0][0 0 1 0][1 0 0 0][0 1 0 0][1 0 0 0]

Crossrefs

Cf. A006052, A007016, A049311, A054976, A057151, A104602, A120732, A319056, A321717, A321723, A321732, A321735, A321736, A321739.

Programs

  • Mathematica
    prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Extensions

a(7)-a(14) from Lars Blomberg, May 23 2019

A230879 Number of 2-packed n X n matrices.

Original entry on oeis.org

1, 2, 56, 16064, 39156608, 813732073472, 147662286695991296, 237776857718965784182784, 3425329990022686416530808209408, 443021337239562368918979788606843912192, 515203019085226443540506018909263027730003787776
Offset: 0

Views

Author

N. J. A. Sloane, Nov 09 2013

Keywords

Comments

A k-packed matrix of size n X n is a matrix with entries in the alphabet A_k = {0,1, ..., k} such that each row and each column contains at least one nonzero entry.

Links

Crossrefs

Row sums of A230878.
Cf. A048291, A055599, A230880, A104602.

Programs

  • Mathematica
    p[k_, n_] := Sum[(-1)^(i + j)*Binomial[n, i]*Binomial[n, j]*(k + 1)^(i*j), {i, 0, n}, {j, 0, n}];
a[n_] := p[2, n];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
  • PARI
    \\ here p(k,n) is number of k-packed matrices of size n.
p(k,n)={sum(i=0, n, sum(j=0, n, (-1)^(i+j) * binomial(n,i) * binomial(n,j) * (k+1)^(i*j)))}
a(n) = p(2,n); \\ Andrew Howroyd, Sep 20 2017

Formula

Cheballah et al. give an explicit formula.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * binomial(n,i) * binomial(n,j) * 3^(i*j). - Andrew Howroyd, Sep 20 2017

Extensions

Terms a(7) and beyond from Andrew Howroyd, Sep 20 2017

A230880 Number of 2-packed matrices with exactly n nonzero entries.

Original entry on oeis.org

1, 2, 8, 80, 1120, 20544, 463744, 12422656, 384947200, 13541822464, 533049493504, 23210958688256, 1107652218822656, 57482801016422400, 3223015475535380480, 194157345516262588416, 12505948470244176953344, 857670052436844788318208, 62395270194815987194789888
Offset: 0

Views

Author

N. J. A. Sloane, Nov 09 2013

Keywords

Comments

A k-packed matrix of size n X n is a matrix with entries in the alphabet A_k = {0,1, ..., k} such that each row and each column contains at least one nonzero entry.

Links

Crossrefs

Cf. A048291, A055599, A230878, A230879, A104602.

Programs

  • Mathematica
    b[n_] := Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}]/n!;
a[n_] := 2^n*b[n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
  • PARI
    \\ here b(n) is A104602.
b(n) = {sum(m=0, n, sum(k=0, n, stirling(n,k,1) * m!^2 * stirling(k,m,2)^2)) / n!}
a(n) = 2^n * b(n); \\ Andrew Howroyd, Sep 20 2017

Formula

Cheballah et al. give an explicit formula.
From Andrew Howroyd, Sep 20 2017: (Start)
a(n) = Sum_{r=1..n} Sum_{i=0..r} Sum_{j=0..r} (-1)^(i+j) * binomial(r,i) * binomial(r,j) * binomial(i*j,n) * 2^n.
a(n) = 2^n * A104602(n).
(End)

Extensions

Terms a(9) and beyond from Andrew Howroyd, Sep 20 2017

A365961 Number of (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums.

Original entry on oeis.org

1, 1, 4, 19, 127, 967, 9063, 94595, 1139708, 15118010, 223571836, 3597458356, 63233950081, 1197193320701, 24418765771835, 532015160784016, 12363381055074017, 304754656068754421, 7952728315095555279, 218848562411197549582, 6338152295627215890669, 192627799720153909693048
Offset: 0

Views

Author

Ludovic Schwob, Sep 23 2023

Keywords

Comments

Let f(n) = number of ordered coprime factorizations of n (A325446(n)); a(n) = sum of f(k) over all terms k in A025487 that have n factors.

Examples

			The a(3) = 19 matrices:
  [1 1 1]
.
  [1 1] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1 0] [0 1] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [0 1] [1 0] [0 1] [1 0 0] [1 0 0] [0 1] [1 0]
  [1] [1 0] [0 1] [0 1] [1 0] [0 1 0] [0 0 1] [1 0] [0 1]
  [1] [0 1] [1 0] [1 0] [0 1] [0 0 1] [0 1 0] [1 0] [0 1]
.
  [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Links

Crossrefs

Cf. A068313, A035341, A321735, A101370, A321733, A104602, A116540.

Programs

  • PARI
    R(n,k)={Vec(-1 + 1/prod(j=1, k, 1 - binomial(k,j)*x^j + O(x*x^n)))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023

Extensions

Terms a(13) and beyond from Andrew Howroyd, Sep 23 2023

A321584 Number of connected (0,1)-matrices with n ones and no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 6, 27, 159, 1154, 9968, 99861, 1138234, 14544650, 205927012, 3199714508, 54131864317, 990455375968, 19488387266842, 410328328297512, 9205128127109576, 219191041679766542, 5521387415218119528, 146689867860276432637, 4099255234885039058842, 120199458455807733040338
Offset: 0

Views

Author

Gus Wiseman, Nov 13 2018

Keywords

Comments

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

Examples

			The a(4) = 27 matrices:
  [1111]
.
  [111][111][111][11][110][110][101][101][100][011][011][010][001]
  [100][010][001][11][101][011][110][011][111][110][101][111][111]
.
  [11][11][11][11][10][10][10][10][01][01][01][01]
  [10][10][01][01][11][11][10][01][11][11][10][01]
  [10][01][10][01][10][01][11][11][10][01][11][11]
.
  [1]
  [1]
  [1]
  [1]

Links

Crossrefs

Cf. A007716, A007718, A049311, A056156, A101370, A104602, A120733, A283877, A319557, A319647, A319616-A319629, A321585.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}] (* Mathematica 7.0+ *)
  • PARI
    NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n, sum(k=1, n, (-1)^(n-k)*binomial(n,k)*C[k])))}
ConnectedMats(M)={my([m,n]=matsize(M), R=matrix(m,n)); for(m=1, m, for(n=1, n, R[m,n] = M[m,n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1,i-1)*binomial(n,j)*R[i,j]*M[m-i,n-j])))); R}
seq(n)={my(M=matrix(n,n,i,j,sum(k=1, n, binomial(i*j,k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~)) ))))} \\ Andrew Howroyd, Jan 17 2024

Extensions

a(7) onwards from Andrew Howroyd, Jan 17 2024
Previous Showing 11-20 of 20 results.

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