A321718 -id:A321718 - cp's OEIS frontend

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

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

A321717 Number of non-normal (0,1) semi-magic rectangles with sum of all entries equal to n.

Original entry on oeis.org

1, 1, 4, 8, 39, 122, 950, 5042, 45594, 366243, 3858148, 39916802, 494852628, 6227020802, 88543569808, 1308012219556, 21086562956045, 355687428096002, 6427672041650478, 121645100408832002, 2437655776358606198, 51091307191310604724, 1125098543553717372868, 25852016738884976640002, 620752122372339473623314, 15511210044577707470250243

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

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

Rectangles must be of size k X m where k and m are divisors of n and k*m >= n. This implies that a(p) = p! + 2 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. There are no 1 X 1 rectangle that satisfies the condition. The 1 X p and p X 1 rectangles are [1....1] and its transpose, the p X p rectangle are necessarily permutation matrices and there are p! permutation matrices of size p X p. It also shows that a(n) >= n! + 2 for n > 1. - Chai Wah Wu, Jan 13 2019

			The a(3) = 8 semi-magic rectangles:
  [1 1 1]
.
  [1] [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Max Alekseyev, Table of n, a(n) for n = 0..71
Wikipedia, Magic square

Cf. A006052, A007016, A068313, A101370, A120733, A271103, A319056.

Cf. A321718, A321719, A321720, A321721, A321722, A321723, A321724, A321725.

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/@#]==Range[Max@@Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! + 2 for p prime. a(n) >= n! + 2 for n > 1. - Chai Wah Wu, Jan 13 2019

a(7) from Chai Wah Wu, Jan 13 2019
a(8)-a(13) from Chai Wah Wu, Jan 14 2019
a(14)-a(15) from Chai Wah Wu, Jan 15 2019
a(16)-a(19) from Chai Wah Wu, Jan 16 2019
Terms a(20) onward from Max Alekseyev, Dec 04 2024

A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 10, 7, 12, 2, 38, 2, 21, 46, 72, 2, 162, 2, 420, 415, 64, 2, 4987, 1858, 110, 9336, 45456, 2, 136018, 2, 1014658, 406578, 308, 3996977, 34937078, 2, 502, 28010167, 1530292965, 2, 508164038, 2, 54902992348, 51712929897, 1269, 2, 3217847072904, 8597641914, 9168720349613

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d, for some d|n.

			Non-isomorphic representatives of the a(2) = 2 through a(6) = 7 multiset partitions:
  {{11}}   {{111}}     {{1111}}       {{11111}}         {{111111}}
  {{1}{2}} {{1}{2}{3}} {{11}{22}}     {{1}{2}{3}{4}{5}} {{111}{222}}
                       {{12}{12}}                       {{112}{122}}
                       {{1}{2}{3}{4}}                   {{11}{22}{33}}
                                                        {{11}{23}{23}}
                                                        {{12}{13}{23}}
                                                        {{1}{2}{3}{4}{5}{6}}
Inequivalent representatives of the a(6) = 7 matrices:
  [6]
.
  [3 0] [2 1]
  [0 3] [1 2]
.
  [2 0 0] [2 0 0] [1 1 0]
  [0 2 0] [0 1 1] [1 0 1]
  [0 0 2] [0 1 1] [0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]
Inequivalent representatives of the a(9) = 7 matrices:
  [9]
.
  [3 0 0] [3 0 0] [2 1 0] [2 1 0] [1 1 1]
  [0 3 0] [0 2 1] [1 1 1] [1 0 2] [1 1 1]
  [0 0 3] [0 1 2] [0 1 2] [0 2 1] [1 1 1]
.
  [1 0 0 0 0 0 0 0 0]
  [0 1 0 0 0 0 0 0 0]
  [0 0 1 0 0 0 0 0 0]
  [0 0 0 1 0 0 0 0 0]
  [0 0 0 0 1 0 0 0 0]
  [0 0 0 0 0 1 0 0 0]
  [0 0 0 0 0 0 1 0 0]
  [0 0 0 0 0 0 0 1 0]
  [0 0 0 0 0 0 0 0 1]

Wikipedia, Magic square

Cf. A006052, A007716, A057150, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321722, A321724, A333733.

a(p) = 2 for p prime corresponding to the 1 X 1 square [p] and the permutation matrices of size p X p with partition (1...10...0). - Chai Wah Wu, Jan 16 2019

a(n) = Sum_{d|n} A333733(d,n/d) for n > 0. - Andrew Howroyd, Apr 11 2020

a(11)-a(13) from Chai Wah Wu, Jan 16 2019
a(14)-a(15) from Chai Wah Wu, Jan 20 2019
Terms a(16) and beyond from Andrew Howroyd, Apr 11 2020

A321722 Number of non-normal magic squares whose entries are nonnegative integers summing to n.

Original entry on oeis.org

1, 1, 1, 1, 10, 21, 97, 657, 5618, 48918, 494530, 5383553, 65112565, 840566081, 11834555867, 176621056393, 2838064404989, 48060623405313

Offset: 0

Gus Wiseman, Nov 18 2018

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.

			The a(4) = 10 magic squares:
  [4]
.
  [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]

Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A120732, A271103, A319056, A319616.

Cf. A321718, A321719, A321720, A321721.

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@@Join[{Tr[prs2mat[#]],Tr[Reverse[prs2mat[#]]]},Total/@prs2mat[#],Total/@Transpose[prs2mat[#]]]]&]],{n,5}]

a(p) = A007016(p) + 1 if p is prime. a(n) >= A007016(n) + 1 for n > 1. - Chai Wah Wu, Jan 15 2019

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

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

A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025

Offset: 0

Gus Wiseman, Jul 13 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(2) = 1 through a(5) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}      {{1,2,3,4,5}}
                      {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                      {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                      {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                       {{1,4,5},{1,2,3,4}}
                                       {{2,3,5},{1,2,3,4}}
                                       {{2,4,5},{1,2,3,5}}
                                       {{3,4,5},{1,2,4,5}}
                                       {{1,5},{2,4},{1,2,3}}
                                       {{2,5},{3,4},{1,2,4}}
                                       {{3,5},{1,2,5},{1,3,4}}
                                       {{4,5},{1,3,5},{2,3,4}}
                                       {{1,4,5},{2,3,5},{1,2,3,4}}

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 14 2019

A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 3, 7, 1, 1, 1, 1, 4, 9, 12, 11, 1, 1, 1, 1, 4, 15, 1, 1, 13, 31, 1, 1, 5, 43, 22, 1, 1, 1, 1, 5, 22, 103, 30, 1, 1, 1, 1, 6, 106, 264, 42, 1, 1, 30, 383, 1, 1, 6, 56, 1, 1, 1, 1, 7, 45, 321, 2804, 1731, 77, 1

Offset: 1

Gus Wiseman, Nov 18 2018

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
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 1
  1 2 1
  1 1
  1 2 3 1
  1 1
  1 3 5 1
  1 5 1
  1 3 7 1
Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
  [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
  [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
  [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
  [    2] [   11] [   11] [   11] [   11] [   11] [   11]

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)
Wikipedia, Magic square

Row sums are A321721.
Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.
Cf. A321718, A321719, A321722, A333733.

T(n,k) = A333733(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Apr 11 2020

a(28)-a(39) from Chai Wah Wu, Jan 16 2019
Terms a(40) and beyond from Andrew Howroyd, Apr 11 2020
Edited by Peter Munn, Mar 05 2025

A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 1, 1, 2, 4, 14, 92, 1320, 73584, 51913039

Offset: 0

Gus Wiseman, Jul 13 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(1) = 1 through a(5) = 14 antichains:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{3},{1,2}}  {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                               {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                               {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                                {{5},{1,4},{2,3}}
                                                {{1,4,5},{1,2,3,4}}
                                                {{2,3,5},{1,2,3,4}}
                                                {{2,4,5},{1,2,3,5}}
                                                {{3,4,5},{1,2,4,5}}
                                                {{1,5},{2,4},{1,2,3}}
                                                {{2,5},{3,4},{1,2,4}}
                                                {{3,5},{1,2,5},{1,3,4}}
                                                {{4,5},{1,3,5},{2,3,4}}
                                                {{1,4,5},{2,3,5},{1,2,3,4}}

The non-covering case is A326574.

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326518, A326534, A326565.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 14 2019

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

Gus Wiseman, Nov 18 2018

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.

			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]

Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A057151, A068313, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321718, A321719, A321720, A321721, A321722.

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}]

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

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

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

A321732 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 3, 11, 53, 317, 2293, 19435, 188851, 2068417, 25203807, 338117445, 4951449055, 78589443061, 1343810727205, 24626270763109, 481489261372381, 10004230113283129, 220125503239710879, 5113204953106107087, 125037079246130168973

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

			The a(3) = 11 matrices:
  [3]
.
  [2 0] [1 1] [1 0] [0 1]
  [0 1] [1 0] [0 2] [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]

Ludovic Schwob, Table of n, a(n) for n = 0..30

Cf. A000700, A000701, A006052, A007016, A120732, A319056, A320451, A321718, A321719, A321722, A321733, A321734, A321735, A321736, A321739.

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/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(7) onwards from Ludovic Schwob, Apr 03 2024

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(4) = 22 antichains:
  {}    {}     {}       {}           {}
  {{}}  {{}}   {{}}     {{}}         {{}}
        {{1}}  {{1}}    {{1}}        {{1}}
               {{2}}    {{2}}        {{2}}
               {{1,2}}  {{3}}        {{3}}
                        {{1,2}}      {{4}}
                        {{1,3}}      {{1,2}}
                        {{2,3}}      {{1,3}}
                        {{1,2,3}}    {{1,4}}
                        {{3},{1,2}}  {{2,3}}
                                     {{2,4}}
                                     {{3,4}}
                                     {{1,2,3}}
                                     {{1,2,4}}
                                     {{1,3,4}}
                                     {{2,3,4}}
                                     {{1,2,3,4}}
                                     {{3},{1,2}}
                                     {{4},{1,3}}
                                     {{1,4},{2,3}}
                                     {{2,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Set partitions with equal block-sums are A035470.
Antichains with different edge-sums are A326030.
MM-numbers of multiset partitions with equal part-sums are A326534.
The covering case is A326566.
Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 13 2019

cp's OEIS Frontend

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

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A321717 Number of non-normal (0,1) semi-magic rectangles with sum of all entries equal to n.

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A321721 Number of non-isomorphic non-normal semi-magic square multiset partitions of weight n.

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A321722 Number of non-normal magic squares whose entries are nonnegative integers summing to n.

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A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

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A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

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A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

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

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