A321719 -id:A321719 - cp's OEIS frontend

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

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

A321718 Number of coupled non-normal semi-magic rectangles with sum of entries equal to n.

Original entry on oeis.org

1, 1, 5, 9, 44, 123, 986, 5043, 45832, 366300, 3862429, 39916803, 495023832, 6227020803, 88549595295, 1308012377572, 21086922542349, 355687428096003, 6427700493998229, 121645100408832003, 2437658338007783347, 51091307195905020227, 1125098837523651728389, 25852016738884976640003, 620752163206546966698620, 15511210044577707492319496

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

A coupled non-normal semi-magic rectangle is a nonnegative integer matrix with equal row sums and equal column sums. The common row sum may be different from the common column sum.

Rectangles must be of size k X m where k and m are divisors of n. This implies that a(p) = p! + 3 for p prime since the only allowable rectangles are of sizes 1 X 1, 1 X p, p X 1 and p X p. The 1 X 1 square is [p], the 1 X p and p X 1 rectangles are [1,...,1] and its transpose and the p X p squares are necessarily permutation matrices and there are p! permutation matrices of size p X p. Also, a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

			The a(3) = 9 coupled semi-magic rectangles:
  [3] [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..69
Wikipedia, Magic square

Cf. A006052, A120733, A271103, A319056, A321654.

Cf. A321717, A321719, A321720, A321721, A321722, A321724, 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/@#]==Range[Max@@Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(p) = p! + 3 for p prime. a(n) >= n! + 3 for n > 1. - Chai Wah Wu, Jan 15 2019

a(7)-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

A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91

Offset: 1

Gus Wiseman, Nov 19 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3

			The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1

Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]

A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 1, 3, 1, 0, 2, 0, 4, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 0, 6, 1, 1, 3, 4, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Nov 19 2018

Row n has length A000041(A056239(n)).

A vertical section is a partial Young diagram with at most one square in each row.

			Triangle begins:
  1
  1
  0  1
  1  1
  0  0  1
  0  2  1
  0  0  0  0  1
  1  3  1
  0  2  0  4  1
  0  0  0  3  1
  0  0  0  0  0  0  1
  0  2  2  5  1
  0  0  0  0  0  0  0  0  0  0  1
  0  0  0  0  0  4  1
  0  0  0  6  0  6  1
  1  3  4  6  1
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
  0  0  4 10  4  8  1
The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
  T(12,7) = 0:
.
  T(12,9) = 2:    1 2   1 2
                  1     2
                  2     1
.
  T(12,10) = 2:   1 2   1 2
                  2     1
                  2     1
.
  T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2
                  3     2     3     1     3
                  3     3     2     3     1
.
  T(12,16) = 1:   1 2
                  3
                  4

Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Table[Length[Select[spsu[ptnverts[y],ptnpos[y]],Sort[Length/@#]==primeMS[k]&]],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]],{n,18}]

A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

First differs from A137944 in lacking 120.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			6480 belongs to the sequence because it is the Heinz number of (3,2,2,2,2,1,1,1,1), which can be arranged in the following ways:
  [1 1 3] [1 2 2] [1 2 2] [1 3 1] [2 1 2] [2 1 2] [2 2 1] [2 2 1] [3 1 1]
  [2 2 1] [1 2 2] [3 1 1] [2 1 2] [1 3 1] [2 1 2] [1 1 3] [2 2 1] [1 2 2]
  [2 2 1] [3 1 1] [1 2 2] [2 1 2] [2 1 2] [1 3 1] [2 2 1] [1 1 3] [1 2 2]

Index entries for sequences related to Heinz numbers

Complement of A323304.
Cf. A006052, A007016, A056239, A112798, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323347, A323348, A323349.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[100],!Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 11, 3, 12, 6, 13, 3, 52, 3, 15, 30, 57, 3, 156, 3, 238, 129, 19, 3, 2221, 6, 21, 415, 3114, 3, 14921, 3, 12853, 1044, 25, 6219, 164743, 3, 27, 2220, 851476, 3, 954088, 3, 434106, 3326714, 31, 3, 24648724, 6, 22309800, 7269, 2737618, 3, 69823653

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

Also the number of non-normal semi-magic rectangles summing to n with no zeros.

Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019

			The a(6) = 11 matrices:
  [6] [3 3] [2 2 2] [1 1 1 1 1 1]
.
  [3] [1 2] [2 1] [1 1 1]
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1]
  [2] [1 1]
  [2] [1 1]
.
  [1]
  [1]
  [1]
  [1]
  [1]
  [1]

Chai Wah Wu, Table of n, a(n) for n = 0..59

Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.

Table[Length[Select[Join@@Table[Partition[cmp,d],{cmp,Join@@Permutations/@IntegerPartitions[n]},{d,Divisors[Length[cmp]]}],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,10}]

a(p) = 3 and a(p^2) = 6 for p prime (see comment). - Chai Wah Wu, Jan 13 2019

a(21)-a(31) from Chai Wah Wu, Jan 13 2019
a(32)-a(53) from Chai Wah Wu, Jan 14 2019
a(54) from Chai Wah Wu, Jan 16 2019

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

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.

Andrew Howroyd, Table of n, a(n) for n = 0..180
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A057151, A068313, A008300, A101370, A104602, A120732, A271103, A319056, A319616.

Cf. A321717, A321719, A321722, A321723.

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}]

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

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

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

cp's OEIS Frontend

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

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A321718 Number of coupled non-normal semi-magic rectangles with sum of 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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A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

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A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

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A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.

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A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

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