A319616 -id:A319616 - cp's OEIS frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

1, 1, 3, 6, 16, 35, 94, 222, 584, 1488, 3977

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 16 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{2},{1,2,2}}
  {{3},{1,2,3}}
  {{1,1},{1,1}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,3},{2,3}}
  {{1},{1},{1,1}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A319616.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319789.

A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

Original entry on oeis.org

1, 1, 3, 29, 1885, 18658259

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated elements).

			Non-isomorphic representatives of the a(3) = 29 covers:
   {{1,3},{2,3}}
   {{1},{2},{3}}
   {{1},{3},{2,3}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{3},{1,3},{2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,3},{2,3}}
   {{2},{3},{1,3},{2,3}}
   {{1},{3},{2,3},{1,2,3}}
   {{2},{3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3}}
   {{2},{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,3},{2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,3},{2,3},{1,2,3}}
   {{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3}}
   {{1},{2},{3},{1,3},{2,3},{1,2,3}}
   {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
   {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Cf. A006126, A007716, A049311, A059201, A283877, A316980, A316983, A318099, A319558, A319559, A319616-A319646, A300913.

a(5) from Max Alekseyev, Jul 13 2022

A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

Original entry on oeis.org

1, 1, 3, 15, 107, 991, 11267, 151721, 2360375, 41650861, 821881709, 17932031225, 428630422697, 11138928977049, 312680873171465, 9428701154866535, 303957777464447449, 10431949496859168189, 379755239311735494421

Offset: 0

Vladeta Jovovic, Aug 18 2006

nonn

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 15 matrices:
  [3]
.
  [2 0] [1 1] [1 1] [1 0] [1 0] [0 2] [0 1] [0 1]
  [0 1] [1 0] [0 1] [1 1] [0 2] [1 0] [2 0] [1 1]
.
  [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]
(End)

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A007716, A048291, A054976, A057149, A057150, A057151, A104601, A104602, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[(-1)^(n-k)*StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k,m]^2,{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]]; Table[Length[Select[multsubs[Tuples[Range[n],2],n],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A048144(k).
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^n.
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.4670932578797312973586879293426... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 2^(log(2)/2-2) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} (1-x)^n * (1 - (1-x)^n)^n. - Paul D. Hanna, Mar 26 2018

A321405 Number of non-isomorphic self-dual set systems of weight n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 6, 9, 16, 28, 47

Offset: 0

Gus Wiseman, Nov 15 2018

Also the number of (0,1) symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 16 set systems:
  {{1}}  {{1}{2}}  {{2}{12}}    {{1}{3}{23}}    {{2}{13}{23}}
                   {{1}{2}{3}}  {{1}{2}{3}{4}}  {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}
.
  {{12}{13}{23}}        {{13}{23}{123}}          {{1}{13}{14}{234}}
  {{3}{23}{123}}        {{1}{23}{24}{34}}        {{12}{13}{24}{34}}
  {{1}{3}{24}{34}}      {{1}{4}{34}{234}}        {{1}{24}{34}{234}}
  {{2}{4}{12}{34}}      {{2}{13}{24}{34}}        {{2}{14}{34}{234}}
  {{1}{2}{3}{5}{45}}    {{3}{4}{14}{234}}        {{3}{4}{134}{234}}
  {{1}{2}{3}{4}{5}{6}}  {{1}{2}{4}{35}{45}}      {{4}{13}{14}{234}}
                        {{1}{3}{5}{23}{45}}      {{1}{2}{34}{35}{45}}
                        {{1}{2}{3}{4}{6}{56}}    {{1}{2}{5}{45}{345}}
                        {{1}{2}{3}{4}{5}{6}{7}}  {{1}{3}{24}{35}{45}}
                                                 {{1}{4}{5}{25}{345}}
                                                 {{2}{4}{12}{35}{45}}
                                                 {{4}{5}{13}{23}{45}}
                                                 {{1}{2}{3}{5}{46}{56}}
                                                 {{1}{2}{4}{6}{34}{56}}
                                                 {{1}{2}{3}{4}{5}{7}{67}}
                                                 {{1}{2}{3}{4}{5}{6}{7}{8}}

Cf. A000219, A007716, A045778, A049311, A135588, A138178, A283877, A316980, A316983, A319616.

Cf. A320796, A320797, A321401, A321403, A321406.

A319719 Number of non-isomorphic connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 10, 14, 48, 95, 305, 822, 2615

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices. Connected antichains are also called clutters.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 10 connected antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}
   {{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001055, A001970, A007716, A007718, A056156, A096827, A253249, A285573, A293994, A318099, A319557, A319616-A319646, A319721.

A321719 Number of non-normal semi-magic squares with sum of entries equal to n.

Original entry on oeis.org

1, 1, 3, 7, 28, 121, 746, 5041, 40608, 362936, 3635017, 39916801, 479206146, 6227020801, 87187426839, 1307674521272, 20923334906117, 355687428096001, 6402415241245577, 121645100408832001, 2432905938909013343, 51090942176372298027, 1124001180562929946213

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

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.

Squares must be of size k X k where k is a divisor of n. This implies that a(p) = p! + 1 for p prime since the only allowable squares are of sizes 1 X 1 and p X p. The 1 X 1 square is [p], the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 1 for n > 1. - Chai Wah Wu, Jan 13 2019

			The a(3) = 7 semi-magic squares:
  [3]
.
  [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]

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A120732, A120733, A257493, A271103, A319056, A319616.

Cf. A321717, A321718, A321720, A321721, A321722, A321723, A321724.

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[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! + 1 for p prime and a(n) >= n! + 1 for n > 1 (see comment above). - Chai Wah Wu, Jan 13 2019

a(n) = Sum_{d|n} A257493(d, n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(7) from Chai Wah Wu, Jan 13 2019
a(6) corrected and a(8)-a(15) added by Chai Wah Wu, Jan 14 2019
a(16)-a(19) from Chai Wah Wu, Jan 16 2019
Terms a(20) and beyond from Andrew Howroyd, Apr 11 2020

A319765 Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 156, 358, 792, 1821

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1},{1},{1,1}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319766, A319767, A319768, A319769, A319773, A319774.

A104602 Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.

Original entry on oeis.org

1, 1, 2, 10, 70, 642, 7246, 97052, 1503700, 26448872, 520556146, 11333475922, 270422904986, 7016943483450, 196717253145470, 5925211960335162, 190825629733950454, 6543503207678564364, 238019066600097607402, 9153956822981328930170, 371126108428565106918404

Offset: 0

Ralf Stephan, Mar 27 2005

nonn

Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns, up to row and column permutation, is A057151(n). - Vladeta Jovovic, Mar 25 2006

			From _Gus Wiseman_, Nov 14 2018: (Start)
The a(3) = 10 matrices:
  [1 1] [1 1] [1 0] [0 1]
  [1 0] [0 1] [1 1] [1 1]
.
  [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]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..400
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math/0503436 [math.CO], 2005.

Row sums of triangle A104601.

Cf. A048291, A049311, A054976, A057150, A057151, A101370, A120732, A120733, A138178, A316983, A319616.

Table[1/n!*Sum[StirlingS1[n,k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}],{k,0,n}],{n,1,20}] (* Vaclav Kotesovec, May 07 2014 *)
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],Union[First/@#]==Union[Last/@#]==Range[Max@@First/@#]&]],{n,5}] (* Gus Wiseman, Nov 14 2018 *)

a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k)*A048144(k). - Vladeta Jovovic, Mar 25 2006
G.f.: Sum_{n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.28889864564457451375789435201798... . - Vaclav Kotesovec, May 07 2014
In closed form, c = 1 / (log(2) * 2^(log(2)/2+2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (1+x)^(n*(n+1)). - Paul D. Hanna, Mar 26 2018

More terms from Vladeta Jovovic, Mar 25 2006

a(0)=1 prepended by Alois P. Heinz, Jan 14 2015

A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

Original entry on oeis.org

1, 3, 17, 179, 3835, 200082, 29610804, 13702979132, 20677458750966, 103609939177198046, 1745061194503344181714, 99860890306900024150675406, 19611238933283757244479826044874, 13340750149227624084760722122669739026, 31706433098827528779057124372265863803044450

Offset: 1

Vladeta Jovovic, May 27 2000

Also the number of non-isomorphic set multipartitions (multisets of sets) with n parts and n vertices. - Gus Wiseman, Nov 18 2018

			From _Gus Wiseman_, Nov 18 2018: (Start)
Inequivalent representatives of the a(3) = 17 matrices:
  100 100 100 100 100 010 010 001 001 001 001 110 101 101 011 011 111
  100 010 001 011 011 001 101 001 101 011 111 101 011 011 011 111 111
  011 001 011 011 111 111 011 111 011 111 111 011 011 111 111 111 111
Non-isomorphic representatives of the a(1) = 1 through a(3) = 17 set multipartitions:
  {{1}}  {{1},{2}}      {{1},{2},{3}}
         {{2},{1,2}}    {{1},{1},{2,3}}
         {{1,2},{1,2}}  {{1},{3},{2,3}}
                        {{1},{2,3},{2,3}}
                        {{2},{1,3},{2,3}}
                        {{2},{3},{1,2,3}}
                        {{3},{1,3},{2,3}}
                        {{3},{3},{1,2,3}}
                        {{1,2},{1,3},{2,3}}
                        {{1},{2,3},{1,2,3}}
                        {{1,3},{2,3},{2,3}}
                        {{3},{2,3},{1,2,3}}
                        {{1,3},{2,3},{1,2,3}}
                        {{2,3},{2,3},{1,2,3}}
                        {{3},{1,2,3},{1,2,3}}
                        {{2,3},{1,2,3},{1,2,3}}
                        {{1,2,3},{1,2,3},{1,2,3}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50

Column sums of A057150.

Cf. A007716, A048291 (labeled), A049311, A057149, A057151, A104601, A104602, A319616, A320808.

A002724 = Cases[Import["https://oeis.org/A002724/b002724.txt", "Table"], {, }][[All, 2]];
A002725 = Cases[Import["https://oeis.org/A002725/b002725.txt", "Table"], {, }][[All, 2]];
a[n_] := A002724[[n + 1]] - 2 A002725[[n]] + A002724[[n]];
a /@ Range[1, 13] (* Jean-François Alcover, Sep 14 2019 *)

a(n) = A002724(n) - 2*A002725(n-1) + A002724(n-1).

More terms from David Wasserman, Mar 06 2002

Terms a(14) and beyond from Andrew Howroyd, Apr 11 2020

A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 3, 1, 1, 7, 14, 10, 3, 1, 1, 9, 23, 24, 11, 3, 1, 1, 12, 39, 53, 34, 12, 3, 1, 1, 14, 61, 102, 86, 39, 12, 3, 1, 1, 17, 90, 193, 201, 117, 42, 12, 3, 1, 1, 20, 129, 340, 434, 310, 136, 43, 12, 3, 1, 1, 24, 184, 584, 902, 778, 412, 149, 44, 12, 3, 1

Offset: 1

Gus Wiseman, Nov 02 2018

Also the number of nonnegative integer k X k symmetric matrices with sum of elements equal to n and no zero rows or columns, up to row and column permutations.
The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
   1
   1   1
   1   2   1
   1   4   3   1
   1   5   7   3   1
   1   7  14  10   3   1
   1   9  23  24  11   3   1
   1  12  39  53  34  12   3   1
   1  14  61 102  86  39  12   3   1
   1  17  90 193 201 117  42  12   3   1
Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):
1: {{1}}
.
2: {{11}}  {{1}{2}}
.
3: {{111}}  {{1}{22}}  {{1}{2}{3}}
.           {{2}{12}}
.
4: {{1111}}  {{11}{22}}  {{1}{1}{23}}  {{1}{2}{3}{4}}
.            {{12}{12}}  {{1}{2}{33}}
.            {{1}{222}}  {{1}{3}{23}}
.            {{2}{122}}
.
5: {{11111}}  {{11}{122}}  {{1}{22}{33}}  {{1}{2}{2}{34}}  {{1}{2}{3}{4}{5}}
.             {{11}{222}}  {{1}{23}{23}}  {{1}{2}{3}{44}}
.             {{12}{122}}  {{1}{2}{333}}  {{1}{2}{4}{34}}
.             {{1}{2222}}  {{1}{3}{233}}
.             {{2}{1222}}  {{2}{12}{33}}
.                          {{2}{13}{23}}
.                          {{3}{3}{123}}

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A316983.

Cf. A000219, A007716, A316980, A317533, A318805, A319560, A319616, A319721, A320797-A320813.

row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

T(n,k) = A318805(k,n) - A318805(k-1,n). - Andrew Howroyd, Jan 16 2024

a(56) onwards from Andrew Howroyd, Jan 16 2024

cp's OEIS Frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

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A319637 Number of non-isomorphic T_0-covers of n vertices by distinct sets.

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A120732 Number of square matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

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A321405 Number of non-isomorphic self-dual set systems of weight n.

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A319719 Number of non-isomorphic connected antichains of multisets of weight n.

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A321719 Number of non-normal semi-magic squares with sum of entries equal to n.

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A319765 Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.

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A104602 Number of square (0,1)-matrices with exactly n entries equal to 1 and no zero row or columns.

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A054976 Number of binary n X n matrices with no zero rows or columns, up to row and column permutation.

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