A326520 -id:A326520 - cp's OEIS frontend

A326518 Number of normal multiset partitions of weight n where every part has the same sum.

1, 1, 3, 7, 15, 31, 75, 169, 445, 1199, 3471

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(4) = 15 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
             {{1},{1}}  {{1,2,2}}      {{1,1,2,2}}
                        {{1,2,3}}      {{1,1,2,3}}
                        {{2},{1,1}}    {{1,2,2,2}}
                        {{3},{1,2}}    {{1,2,2,3}}
                        {{1},{1},{1}}  {{1,2,3,3}}
                                       {{1,2,3,4}}
                                       {{1,1},{1,1}}
                                       {{1,2},{1,2}}
                                       {{1,3},{2,2}}
                                       {{1,4},{2,3}}
                                       {{2},{2},{1,1}}
                                       {{3},{3},{1,2}}
                                       {{1},{1},{1},{1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A035470, A038041, A255906, A317583, A321455, A326517, A326519, A326520, A326521, A326534.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Total/@#&]],{n,0,5}]

a(10) from Robert Price, Apr 04 2025

A326519 Number of normal multiset partitions of weight n where each part has a different sum.

Original entry on oeis.org

1, 1, 3, 11, 51, 259, 1461, 9133, 62348, 459547, 3632419

Offset: 0

Gus Wiseman, Jul 12 2019

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 11 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{2}}  {{1,2,2}}
                        {{1,2,3}}
                        {{1},{1,1}}
                        {{1},{1,2}}
                        {{1},{2,2}}
                        {{1},{2,3}}
                        {{2},{1,2}}
                        {{2},{1,3}}
                        {{1},{2},{3}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A275780, A317583, A321469, A326517, A326518, A326520, A326521, A326535.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Total/@#&]],{n,0,5}]

a(8)-a(10) from Robert Price, Apr 03 2025

A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

			The a(3) = 8 multiset partitions:
  {{1,1,1}}
  {{1,1,2}}
  {{1,2,2}}
  {{1,2,3}}
  {{1},{1},{1}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521, A331315.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Length/@#&]],{n,8}]

\\ here U(n,m) gives number for m blocks of size n.
U(n,m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )}
a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018

a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020

Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018

A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 7, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The a(900) = 9 factorizations:
  (3*3*10*10),
  (3*3*100), (3*10*30), (9*10*10),
  (3*300), (9*100), (10*90), (30*30),
  (900).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326512, A326514, A326516, A326520, A326536.

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]]}]];
Table[Length[Select[facs[n],SameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
has_same_average_of_pis(facs) = if(!#facs, 1, my(avg=0); for(i=1,#facs,if(!avg, avg=avgpis(facs[i]), if(avg!=avgpis(facs[i]), return(0)))); (1));
A326515(n, m=n, facs=List([])) = if(1==n, has_same_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326515(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A326517 Number of normal multiset partitions of weight n where each part has a different size.

Original entry on oeis.org

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

A326512 Number of set partitions of {1..n} where every block has the same average.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 18, 16, 75, 64, 405, 302, 2581, 1693, 19872, 11295, 175807, 87524, 1851135, 787515, 21909766, 8185713, 298698113, 96514608, 4538610230, 1285072142

Offset: 0

Gus Wiseman, Jul 11 2019

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024

			The a(1) = 1 through a(7) = 18 set partitions:
  {1}  {12}  {123}    {1234}    {12345}      {123456}      {1234567}
             {13}{2}  {14}{23}  {1245}{3}    {1256}{34}    {123567}{4}
                                {135}{24}    {1346}{25}    {12467}{35}
                                {15}{234}    {16}{2345}    {1267}{345}
                                {15}{24}{3}  {16}{25}{34}  {13457}{26}
                                                           {1357}{246}
                                                           {1456}{237}
                                                           {147}{2356}
                                                           {156}{2347}
                                                           {17}{23456}
                                                           {1267}{35}{4}
                                                           {1357}{26}{4}
                                                           {147}{26}{35}
                                                           {156}{237}{4}
                                                           {17}{2356}{4}
                                                           {17}{246}{35}
                                                           {17}{26}{345}
                                                           {17}{26}{35}{4}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A070925, A275780, A306017, A306021, A319169, A326513, A326515, A326520, A326536.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],SameQ@@Mean/@#&]],{n,0,8}]

a(12)-a(15) from Alois P. Heinz, Jul 12 2019

a(16)-a(26) from Christian Sievers, Aug 22 2024

A326521 Number of normal multiset partitions of weight n where each part has a different average.

Original entry on oeis.org

1, 1, 3, 11, 49, 251, 1418, 8904

Offset: 0

Gus Wiseman, Jul 12 2019

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 11 normal multiset partitions where each part has a different average:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{2}}  {{1,2,2}}
                        {{1,2,3}}
                        {{1},{1,2}}
                        {{1},{2,2}}
                        {{1},{2,3}}
                        {{2},{1,1}}
                        {{2},{1,2}}
                        {{3},{1,2}}
                        {{1},{2},{3}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A051293, A255906, A317583, A326513, A326516, A326517, A326518, A326519, A326520, A326537.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Mean/@#&]],{n,0,5}]

A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 5, 8, 8, 10, 8, 15, 9, 14, 15, 17, 13, 22, 14, 25, 21, 23, 19, 34, 24, 29, 28, 37, 27, 45, 29, 44, 38, 43, 43, 59, 40, 51, 48, 69, 48, 71, 52, 73, 69, 72, 61, 93, 72, 91, 77, 99, 78, 105, 95, 119, 95, 113, 96, 146, 107, 126, 123, 151, 130

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 7 multiset partitions:
  {1} {11}   {111}     {1111}       {11111}         {111111}
      {1}{1} {2}{11}   {11}{11}     {2}{11}{11}     {111}{111}
             {1}{1}{1} {2}{2}{11}   {2}{2}{2}{11}   {22}{1111}
                       {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
                                                    {2}{2}{11}{11}
                                                    {2}{2}{2}{2}{11}
                                                    {1}{1}{1}{1}{1}{1}
The a(1) = 1 through a(7) = 5 factorizations:
  2  4    8      16       32         64           128
     2*2  3*4    4*4      3*4*4      8*8          3*4*4*4
          2*2*2  3*3*4    3*3*3*4    9*16         3*3*3*4*4
                 2*2*2*2  2*2*2*2*2  4*4*4        3*3*3*3*3*4
                                     3*3*4*4      2*2*2*2*2*2*2
                                     3*3*3*3*4
                                     2*2*2*2*2*2

Christian Sievers, Table of n, a(n) for n = 0..25000

Without a common sum we have A055887.
Twice-partitions of this type are counted by A279789.
Without constant blocks we have A326518.
For distinct block-sums and strict blocks we have A381718.
Factorizations of this type are counted by A381995.
For distinct instead of equal block-sums we have A382203.
For strict instead of constant blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A255906 counts normal multiset partitions, row sums of A317532.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A304969, A356945.
Set multipartitions: A116540, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

h(s,x)=my(t=0,p=1,k=1);while(s%k==0,p*=1/(1-x^(s/k))-1;t+=p;k+=1);t
lista(n)=Vec(1+sum(s=1,n,h(s,x+O(x*x^n)))) \\ Christian Sievers, Apr 05 2025

G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - Christian Sievers, Apr 05 2025

Terms a(16) and beyond from Christian Sievers, Apr 04 2025

A336137 Number of set partitions of the binary indices of n with equal block-sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 7, 59, 119, 367, 127:
  {123}    {12456}      {123567}      {1234679}    {1234567}
  {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
           {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
                        {17}{26}{35}  {1267}{349}  {1346}{257}
                                      {169}{2347}  {167}{2345}
                                                   {16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
  {{2,7},{3,6},{4,5},{9}}
  {{2,4,6},{3,9},{5,7}}
  {{2,7,9},{3,4,5,6}}
  {{2,3,4,9},{5,6,7}}
  {{2,3,6,7},{4,5,9}}
  {{2,4,5,7},{3,6,9}}
  {{2,3,4,5,6,7,9}}
so a(382) = 7.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

A382429 Number of normal multiset partitions of weight n into sets with a common sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 13, 26, 57, 113, 283, 854, 2401, 6998, 24072, 85061, 308956, 1190518, 4770078, 19949106, 87059592

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(6) = 13 partitions:
  {1} {12}   {123}     {1234}       {12345}         {123456}
      {1}{1} {3}{12}   {12}{12}     {24}{123}       {123}{123}
             {1}{1}{1} {14}{23}     {34}{124}       {125}{134}
                       {3}{3}{12}   {3}{12}{12}     {135}{234}
                       {1}{1}{1}{1} {5}{14}{23}     {145}{235}
                                    {3}{3}{3}{12}   {12}{12}{12}
                                    {1}{1}{1}{1}{1} {14}{14}{23}
                                                    {14}{23}{23}
                                                    {16}{25}{34}
                                                    {3}{3}{12}{12}
                                                    {5}{5}{14}{23}
                                                    {3}{3}{3}{3}{12}
                                                    {1}{1}{1}{1}{1}{1}
The corresponding factorizations:
  2  6    30     210      2310       30030
     2*2  5*6    6*6      21*30      30*30
          2*2*2  14*15    35*42      6*6*6
                 5*5*6    5*6*6      66*70
                 2*2*2*2  5*5*5*6    110*105
                          11*14*15   154*165
                          2*2*2*2*2  5*5*6*6
                                     14*14*15
                                     14*15*15
                                     26*33*35
                                     5*5*5*5*6
                                     11*11*14*15
                                     2*2*2*2*2*2

Without the common sum we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279788.
For common sizes instead of sums we have A317583.
Without strict blocks we have A326518, non-strict blocks A326517.
For a common length instead of sum we have A331638.
For distinct instead of equal block-sums we have A381718.
Factorizations of this type are counted by A382080.
For distinct block-sums and constant blocks we have A382203.
For constant instead of strict blocks we have A382204.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A255906, A304969, A317532.
Set multipartitions: A089259, A116539, A270995, A296119, A318360.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Constant blocks with distinct sums: A381635, A381636, A381716.
Cf. A000110, A034691, A038041, A050320, A255903, A326520, A381633, A381996, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(11) from Robert Price, Mar 30 2025

a(12)-a(20) from Christian Sievers, Apr 06 2025

cp's OEIS Frontend

A326518 Number of normal multiset partitions of weight n where every part has the same sum.

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A326519 Number of normal multiset partitions of weight n where each part has a different sum.

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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

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A326515 Number of factorizations of n into factors > 1 where every factor has the same average of prime indices.

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A326517 Number of normal multiset partitions of weight n where each part has a different size.

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A326512 Number of set partitions of {1..n} where every block has the same average.

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A326521 Number of normal multiset partitions of weight n where each part has a different average.

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