A120733 -id:A120733 - cp's OEIS frontend

A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

1, 1, 1, 5, 5, 9, 45, 49, 85, 125, 233, 273, 417, 529, 745, 2573, 2861, 4761, 6837, 10489, 14317, 22637, 28289, 40041, 52041, 70177, 88561, 117605, 234773, 274761, 407469, 553681, 792613, 1052525, 1493033, 1959009, 3135537, 3904129, 5475673, 7173725, 9853325

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

			The a(5) = 9 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [3] [2] [1]
  [1] [2] [3] [4]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000005, A000219, A007716, A008289, A114736, A117433, A120733, A321645, A321653, A321655, A321659, A321661, A321662.

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/@#],UnsameQ@@Join@@prs2mat[#]]&]],{n,5}]

seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, (n-1)!*numdiv(n-1) + n!*(numdiv(n) - 2)))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} (k!*A000005(k) + (k+1)!*(A000005(k+1) - 2))*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 50, 69, 125, 177, 301, 440, 717, 1055, 1675, 2471, 3835, 5660, 8627, 12697, 19095, 27978, 41581, 60650, 89244, 129490, 188925, 272676, 394809, 566882, 815191, 1164510, 1664295, 2365698, 3361844, 4756030, 6723280, 9468138, 13319299

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

First differs from A304967 at a(10) = 50, A304967(10) = 49.
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 positive integer matrices with only two columns and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{1222}}    {{12222}}    {{112222}}      {{1122222}}
                   {{12}{12}}  {{12}{122}}  {{122222}}      {{1222222}}
                                            {{112}{122}}    {{112}{1222}}
                                            {{12}{1122}}    {{12}{11222}}
                                            {{12}{1222}}    {{12}{12222}}
                                            {{122}{122}}    {{122}{1122}}
                                            {{12}{12}{12}}  {{122}{1222}}
                                                            {{12}{12}{122}}
Inequivalent representatives of the a(8) = 20 matrices:
  [4 4] [3 5] [2 6] [1 7]
.
  [1 1] [1 1] [1 1] [2 1] [2 1] [1 2] [1 2] [3 1] [2 2] [2 2] [1 3]
  [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]
.
  [1 1] [1 1] [1 1] [1 1]
  [1 1] [1 1] [2 1] [1 2]
  [2 2] [1 3] [1 2] [1 2]
.
  [1 1]
  [1 1]
  [1 1]
  [1 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1,(EulerT(vector(n, k, k-1)) + EulerT(vector(n, k, if(k%2, 0, (k+2)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

Original entry on oeis.org

1, 1, 4, 7, 19, 35, 80, 149, 307, 566, 1092, 1974, 3643, 6447, 11498, 19947, 34636, 58974, 100182, 167713, 279659, 461056, 756562, 1230104, 1990255, 3195471, 5105540, 8103722, 12801925, 20107448, 31439978, 48907179, 75755094, 116797754, 179354540, 274253042

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

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 matrices with only one or two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:
  {{1}}  {{11}}    {{111}}      {{1111}}
         {{12}}    {{122}}      {{1122}}
         {{1}{1}}  {{1}{11}}    {{1222}}
         {{1}{2}}  {{1}{22}}    {{1}{111}}
                   {{2}{12}}    {{11}{11}}
                   {{1}{1}{1}}  {{1}{122}}
                   {{1}{2}{2}}  {{11}{22}}
                                {{12}{12}}
                                {{1}{222}}
                                {{12}{22}}
                                {{2}{122}}
                                {{1}{1}{11}}
                                {{1}{1}{22}}
                                {{1}{2}{12}}
                                {{1}{2}{22}}
                                {{2}{2}{12}}
                                {{1}{1}{1}{1}}
                                {{1}{1}{2}{2}}
                                {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:
  {{1}}  {{11}}    {{111}}    {{1111}}
         {{12}}    {{122}}    {{1122}}
         {{1}{1}}  {{123}}    {{1222}}
         {{1}{2}}  {{1}{11}}  {{1233}}
                   {{1}{22}}  {{1234}}
                   {{1}{23}}  {{1}{111}}
                   {{2}{12}}  {{11}{11}}
                              {{1}{122}}
                              {{11}{22}}
                              {{12}{12}}
                              {{1}{222}}
                              {{12}{22}}
                              {{1}{233}}
                              {{12}{33}}
                              {{1}{234}}
                              {{12}{34}}
                              {{13}{23}}
                              {{2}{122}}
                              {{3}{123}}
Inequivalent representatives of the a(4) = 19 matrices:
  [4] [2 2] [1 3]
.
  [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]
  [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]
.
  [1] [1 0] [1 0] [1 0] [0 1]
  [1] [1 0] [0 1] [0 1] [0 1]
  [2] [0 2] [1 1] [0 2] [1 1]
.
  [1] [1 0] [1 0]
  [1] [1 0] [0 1]
  [1] [0 1] [0 1]
  [1] [0 1] [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A007716, A052847, A005380, A054974, A005986, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

Original entry on oeis.org

0, 0, 2, 4, 14, 28, 69, 134, 285, 536, 1050, 1918, 3566, 6346, 11363, 19771, 34405, 58677, 99797, 167223, 279032, 460264, 755560, 1228849, 1988680, 3193513, 5103104, 8100712, 12798207, 20102883, 31434374, 48900337, 75746745, 116787611, 179342230, 274238159

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

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 matrices with only two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:
  {{12}}    {{122}}      {{1122}}
  {{1}{2}}  {{1}{22}}    {{1222}}
            {{2}{12}}    {{1}{122}}
            {{1}{2}{2}}  {{11}{22}}
                         {{12}{12}}
                         {{1}{222}}
                         {{12}{22}}
                         {{2}{122}}
                         {{1}{1}{22}}
                         {{1}{2}{12}}
                         {{1}{2}{22}}
                         {{2}{2}{12}}
                         {{1}{1}{2}{2}}
                         {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:
  {{1}{1}}  {{1}{11}}  {{1}{111}}
  {{1}{2}}  {{1}{22}}  {{11}{11}}
            {{1}{23}}  {{1}{122}}
            {{2}{12}}  {{11}{22}}
                       {{12}{12}}
                       {{1}{222}}
                       {{12}{22}}
                       {{1}{233}}
                       {{12}{33}}
                       {{1}{234}}
                       {{12}{34}}
                       {{13}{23}}
                       {{2}{122}}
                       {{3}{123}}
Inequivalent representatives of the a(4) = 14 matrices:
  [2 2] [1 3]
.
  [1 0] [1 0] [0 1] [2 0] [1 1] [1 1]
  [1 2] [0 3] [1 2] [0 2] [1 1] [0 2]
.
  [1 0] [1 0] [1 0] [0 1]
  [1 0] [0 1] [0 1] [0 1]
  [0 2] [1 1] [0 2] [1 1]
.
  [1 0] [1 0]
  [1 0] [0 1]
  [0 1] [0 1]
  [0 1] [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(0, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2 - EulerT(vector(n,k,1)))} \\ Andrew Howroyd, Aug 26 2019

a(n) = A323655(n) - A000041(n). - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A007322 Number of functors of degree n from free Abelian groups to Abelian groups.

Original entry on oeis.org

1, 6, 39, 320, 3281, 40558, 586751, 9719616, 181353777, 3762893750, 85934344775, 2141853777856, 57852105131809, 1683237633305502, 52483648929669119, 1745835287515739328, 61712106494672572641, 2309989101145068446502, 91279147976756195994983

Offset: 1

Don Zagier (don.zagier(AT)mpim-bonn.mpg.de), Apr 11 1994

H. J. Baues, Quadratic functors and metastable homotopy, Jnl. Pure and Applied Algebra, 91 (1994), 49-107.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..400
Vaclav Kotesovec, Asymptotics of the sequence A120733

A120733[n_] := A120733[n] = Sum[2^(-2-r-s)*Binomial[n+r*s-1, n] , {r, 0, Infinity}, {s, 0, Infinity}]; a[n_] := Sum[A120733[k], {k, 1, n}]; Table[Print[an = a[n]]; an, {n, 1, 18}] (* Jean-François Alcover, May 15 2012, after Vladeta Jovovic *)

Binomial transform of A101370. - Vladeta Jovovic, Aug 17 2006
a(n) = (1/n!)*Sum_{k=1..n} (-1)^(n-k)*Stirling1(n+1,k+1)*A000670(k)^2. - Vladeta Jovovic, Aug 17 2006
G.f.: (1/(1-x))*Sum_{m>0,n>0} Sum_{j=1..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^m. - Vladeta Jovovic, Aug 17 2006
Partial sums of A120733. - Vladeta Jovovic, Aug 21 2006
a(n) ~ 2^(log(2)/2-2) * n! / (log(2))^(2*n+2). - Vaclav Kotesovec, May 03 2015

More terms from Vladeta Jovovic, Aug 17 2006

A321410 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 15, 35, 69, 149, 301

Offset: 0

Gus Wiseman, Nov 16 2018

A multiset is aperiodic if its multiplicities are relatively prime.
Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with relatively prime row sums (or column sums) and no row or column having a common divisor > 1.
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(6) = 15 multiset partitions:
  {1}  {1}{2}  {2}{12}    {2}{122}      {12}{122}        {2}{12222}
               {1}{2}{3}  {1}{1}{23}    {2}{1222}        {1}{23}{233}
                          {1}{3}{23}    {1}{23}{23}      {1}{3}{2333}
                          {1}{2}{3}{4}  {1}{3}{233}      {2}{13}{233}
                                        {2}{13}{23}      {3}{23}{123}
                                        {3}{3}{123}      {3}{3}{1233}
                                        {1}{2}{2}{34}    {1}{1}{1}{234}
                                        {1}{2}{4}{34}    {1}{2}{34}{34}
                                        {1}{2}{3}{4}{5}  {1}{2}{4}{344}
                                                         {1}{3}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}

Cf. A000219, A007716, A120733, A138178, A316983, A319616.

Cf. A320796, A320803, A320806, A320809, A320813, A321283, A321408, A321409, A321411.

A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 1, 9, 29, 181, 1285, 10635, 102355, 1118021, 13637175, 184238115, 2727293893, 43920009785, 764389610843, 14297306352937, 286014489487815, 6093615729757841, 137750602009548533, 3293082026520294529, 83006675263513350581, 2200216851785981586729, 61180266502369886181253

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

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

			The a(4) = 29 matrices:
4 31 13
.
3 21 21 20 12 12 11 110 11 110 101 101 1 10 10 02 011 011 01 01
1 10 01 11 10 01 20 101 02 011 110 011 3 21 12 11 110 101 21 12
.
11 11 10 10 01 01
10 01 11 01 11 10
01 10 01 11 10 11

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A007718, A056156, A059201, A120733, A283877, A316980, A319557, A319558, A319559, A319565, A319647, A319616-A319629, A321446, A321515, A369285.

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]}]];
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[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}]

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,wf)={prod(j=1, #q, wf(t*q[j]))-1}
Q(m,n,wf=w->2)={my(s=0); forpart(p=m, s+=(-1)^#p*permcount(p)*exp(-sum(t=1, n, (-1)^t*x^t*K(p,t,wf)/t, O(x*x^n))) ); Vec((-1)^m*serchop(serlaplace(s),1), -n)}
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(R=vectorv(n,m,Q(m,n,w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum( vecsum( Vec( ConnectedMats( Mat(R))))))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 24 2024

A321661 Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 22, 25, 40, 58, 186, 204, 347, 478, 734, 2033, 2402, 3814, 5464, 8142, 11058, 30142, 34437, 55940, 77794, 116954, 156465, 229462, 533612, 640544, 994922, 1397896, 2048316, 2778750, 3987432, 5292293, 11921070, 14076550, 21802928, 29917842, 44080285

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, whose nonzero entries are all distinct.
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(6) = 22 multiset partitions:
  {{1}}  {{11}}  {{111}}    {{1111}}    {{11111}}    {{111111}}
                 {{122}}    {{1222}}    {{11222}}    {{112222}}
                 {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}
                 {{1}{22}}  {{1}{222}}  {{1}{1111}}  {{122333}}
                                        {{11}{111}}  {{1}{11111}}
                                        {{11}{222}}  {{11}{1111}}
                                        {{1}{2222}}  {{1}{11222}}
                                                     {{11}{1222}}
                                                     {{11}{2222}}
                                                     {{112}{222}}
                                                     {{11}{2333}}
                                                     {{1}{22222}}
                                                     {{122}{222}}
                                                     {{1}{22333}}
                                                     {{122}{333}}
                                                     {{2}{11222}}
                                                     {{22}{1222}}
                                                     {{1}{11}{111}}
                                                     {{1}{11}{222}}
                                                     {{1}{22}{222}}
                                                     {{1}{22}{333}}
                                                     {{2}{11}{222}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000219, A007716, A008289, A059201, A059849, A114736, A117433, A120733, A321653, A321659, A321660, A321662.

\\ here b(n) is A059849(n).
b(n)={sum(k=0, n, stirling(n,k,1)*sum(i=0, k, stirling(k,i,2))^2)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p,i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

a(n) = Sum_{k>=1} A059849(k)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

The first term of this sequence absent from A106543 is 144.

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

Index entries for sequences related to Heinz numbers

Complement of A323306.
Cf. A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323302, A323305, 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[2,1000],Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 27, 36, 54, 66, 99, 128, 169, 221, 295, 367, 488, 610, 779, 993, 1253, 1525, 1955, 2426, 2986, 3684, 4563, 5519, 6840, 8298, 10097, 12298, 14874, 17716, 21635, 26002, 31105, 37081, 44581, 52916, 63259, 74852, 88703, 105543, 124752, 145740, 173522, 203999, 239737, 280424, 329929

Offset: 0

Gus Wiseman, Jan 13 2019

nonn

			The a(8) = 17 integer partitions:
  (53), (62), (71),
  (332), (422), (431), (521), (611),
  (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111).

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

A000041(n) = A323347(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(17)-a(53) from Chai Wah Wu, Jan 15 2019

cp's OEIS Frontend

A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

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A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

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A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

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A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

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A007322 Number of functors of degree n from free Abelian groups to Abelian groups.

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A321410 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic multisets whose sizes are relatively prime.

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A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

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A321661 Number of non-isomorphic multiset partitions of weight n where the nonzero entries of the incidence matrix are all distinct.

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