A120733 -id:A120733 - cp's OEIS frontend

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969

Offset: 1

Gus Wiseman, Nov 15 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the column sums are not relatively prime.
Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
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(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,1},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,2}}
                      {{1,2},{2,2}}
                      {{1,2},{3,3}}
                      {{1,2},{3,4}}
                      {{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
  {{1},{1}}  {{1},{1,1}}    {{1,1,2,2}}        {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1},{1,1,1}}      {{1,1},{1,1,1}}
                            {{1,1},{1,1}}      {{1},{1},{1,1,1}}
                            {{1},{1,2,2}}      {{1},{1,1},{1,1}}
                            {{1,1},{2,2}}      {{1},{1},{1},{1,1}}
                            {{1,2},{1,2}}      {{1},{1},{1},{1},{1}}
                            {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}

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

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n));Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 7, 9, 37, 79, 273, 755, 2648, 8432, 29872, 104624, 384759, 1432655, 5502563, 21533141, 86291313, 352654980, 1471073073, 6253397866, 27083003687, 119399628021, 535591458635, 2443030798539, 11326169401988, 53343974825122, 255121588496338

Offset: 0

Gus Wiseman, Nov 29 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n in which every row and column has at least two nonzero entries.

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) = 1 through a(7) = 9 multiset partitions:
  {{1,2},{1,2}}  {{1,2},{1,2,2}}  {{1,1,2},{1,2,2}}    {{1,1,2},{1,2,2,2}}
                                  {{1,2},{1,1,2,2}}    {{1,2},{1,1,2,2,2}}
                                  {{1,2},{1,2,2,2}}    {{1,2},{1,2,2,2,2}}
                                  {{1,2,2},{1,2,2}}    {{1,2,2},{1,1,2,2}}
                                  {{1,2,3},{1,2,3}}    {{1,2,2},{1,2,2,2}}
                                  {{1,2},{1,2},{1,2}}  {{1,2,3},{1,2,3,3}}
                                  {{1,2},{1,3},{2,3}}  {{1,2},{1,2},{1,2,2}}
                                                       {{1,2},{1,3},{2,3,3}}
                                                       {{1,3},{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A369286.

Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320665, A320798, A320801, A320808, A321407.

Vec(G(20,1)) \\ G defined in A369286. - Andrew Howroyd, Jan 28 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A321407 Number of non-isomorphic multiset partitions of weight n with no constant parts.

Original entry on oeis.org

1, 0, 1, 2, 7, 13, 47, 111, 367, 1057, 3474, 11116, 38106, 131235, 470882, 1720959, 6472129, 24860957, 97779665, 392642763, 1610045000, 6732768139, 28699327441, 124600601174, 550684155992, 2476019025827, 11320106871951, 52598300581495, 248265707440448, 1189855827112636, 5787965846277749

Offset: 0

Gus Wiseman, Nov 29 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which every row has at least two nonzero entries.

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(2) = 1 through a(5) = 13 multiset partitions:
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320798, A320801, A320808, A321760.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
S(q, t, k)={sum(j=1, #q, if(t%q[j]==0, q[j]))*vector(k,i,1)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(x*Ser(K(q, t, n\t)-S(q, t, n\t))/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A321653 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 14, 44, 72, 147, 381, 1405

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(5) = 14 matrices:
  [5] [4 1] [3 2]
.
  [4] [4 0] [3 1] [3 1] [3] [3 0] [3 0] [2 2] [2 1] [2 1] [1 2]
  [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1] [2 0]

Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321652, A321654, A321655.

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/@#],OrderedQ[Total/@prs2mat[#],Greater],OrderedQ[Total/@Transpose[prs2mat[#]],Greater]]&]],{n,6}]

A321655 Number of distinct row/column permutations of strict plane partitions of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 33, 53, 77, 225

Offset: 0

Gus Wiseman, Nov 15 2018

			The a(6) = 9 permutations of strict plane partitions:
  [6] [2 4] [4 2] [1 5] [5 1] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [0 1] [1 0] [2 3] [3 2] [2] [4] [0 2] [1 3] [2 0] [3 1]
  [5] [1] [2 3] [3 2] [0 1] [1 0] [4] [2] [1 3] [0 2] [3 1] [2 0]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Cf. A000219, A001970, A007716, A008480, A068313, A114736, A117433, A120733, A321645, A319646, A321647, A321648.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
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/@#],UnsameQ@@Length/@Split[#],OrderedQ[Sort[Map[Last,GatherBy[Sort[Reverse/@#],First],{2}],submultisetQ],submultisetQ],OrderedQ[Sort[Sort/@Map[Last,GatherBy[#,First],{2}],submultisetQ],submultisetQ]]&]],{n,5}]

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

A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

Original entry on oeis.org

1, 1, 5, 19, 107, 573, 4050, 29093, 249301, 2271020, 23378901, 257871081, 3132494380, 40693204728, 572089068459, 8566311524788, 137165829681775, 2327192535461323, 41865158805428687, 793982154675640340, 15863206077534914434, 332606431999260837036, 7309310804287502958322, 167896287022455809865568

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(3) = 19 matrices:
  [3] [2 1] [1 1 1]
.
  [2] [2 0] [1 1] [1 1 0] [1 0 1] [0 1 1]
  [1] [0 1] [1 0] [0 0 1] [0 1 0] [1 0 0]
.
  [1] [1 0] [1 0] [1 0 0] [1 0 0] [0 1] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [1] [1 0] [0 1] [0 1 0] [0 0 1] [1 0] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [1] [0 1] [1 0] [0 0 1] [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..39
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Cf. A000219, A001970, A007716, A068313, A114736, A120733, A319646, A321645, A321653, A321654, A321655.

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/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]]]&]],{n,6}]

Sum of coefficients in the expansions of all homogeneous symmetric functions in terms of monomial symmetric functions. In other words, if Sum_{|y| = n} h(y) = Sum_{|y| = n} c_y * m(y), then a(n) = Sum_{|y| = n} c_y.

a(10) onwards from Ludovic Schwob, Aug 29 2023

A321446 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 2, 10, 72, 624, 6522, 80178, 1129368, 17917032, 316108752, 6138887616, 130120838400, 2989026225696, 73964789192400, 1961487062520720, 55495429438186920, 1668498596700706440, 53122020640948010640, 1785467619718933936560, 63175132023953553400440

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000612, A007716, A049311, A101370, A120733, A135589, A283877, A316980, A319559, A321515, A321586, A321587, A321588, A369285.

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/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]]]&]],{n,6}]

\\ Q(m, n, wf) defined in A321588.
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1 + y^w + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum(vecsum(R)))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 20 2024

A323347 Number of integer partitions of n whose parts can be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 6, 2, 11, 2, 7, 7, 10, 2, 18, 2, 17, 13, 9, 2, 50, 3, 10, 24, 34, 2, 85, 2, 51, 46, 12, 9, 261, 2, 13, 80, 257, 2, 258, 2, 323, 431, 15, 2, 1533, 3, 227, 206, 1165, 2, 971, 483, 2409, 309, 18, 2

Offset: 0

Gus Wiseman, Jan 13 2019

Rectangles must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 2 for p prime, since the only allowable rectangles must be of size 1 X 1 corresponding to the partition (p), or 1 X p or p X 1 corresponding to the partition (1,1,...,1). Similarly, a(p^2) = 3 since the allowable rectangles must be of sizes 1 X 1 (partition (p^2)), 1 X p or p X 1 (partition (p,p,...,p)), 1 X p^2, p^2 X 1 and p X p (partition (1,1,...,1)). - Chai Wah Wu, Jan 14 2019

			The a(8) = 5 integer partitions are (8), (44), (2222), (3311), (11111111).
The a(12) = 11 integer partitions (C = 12):
  (C)
  (66)
  (444)
  (3333)
  (4422)
  (5511)
  (222222)
  (332211)
  (22221111)
  (222111111)
  (111111111111)
For example, the arrangements of (222111111) are:
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

A000041(n) = A323348(n) + a(n).
Cf. A006052, A007016, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323304, A323306, 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/@#&];
Table[Length[Select[IntegerPartitions[n],!Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

a(p) = 2 and a(p^2) = 3 for p prime (see comment). - Chai Wah Wu, Jan 14 2019

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

a(54)-a(59) from Chai Wah Wu, Jan 16 2019

cp's OEIS Frontend

A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

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A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

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A321407 Number of non-isomorphic multiset partitions of weight n with no constant parts.

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A321653 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with strictly decreasing row sums and column sums.

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A321655 Number of distinct row/column permutations of strict plane partitions of n.

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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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A321652 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

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