A275780 -id:A275780 - cp's OEIS frontend

A278339 Number of set partitions of [n] into subsets whose element sums are distinct squares.

1, 1, 0, 0, 1, 0, 1, 0, 1, 23, 9, 41, 248, 277, 1556, 2854, 5233, 20701, 145137, 1626890, 4118910, 9963276, 9260756, 64027363, 365237571, 1002679107, 21594036300, 24465529531, 144914973347, 1921444799766

Offset: 0

Alois P. Heinz, Nov 18 2016

nonn

			a(0) = 1: {}.
a(1) = 1: 1.
a(4) = 1: 1|234.
a(6) = 1: 1|2356|4.
a(8) = 1: 12345678.
a(9) = 23: 12345678|9, 123568|4|79, 1236789|45, 1245789|36, 126|345789, 12589|367|4, 1258|3679|4, 12679|358|4, 1267|3589|4, 1345689|27, 135|246789, 13|24568|79, 13579|268|4, 1357|2689|4, 13678|259|4, 13|259|4678, 13|2689|457, 13|268|4579, 156789|234, 18|2345679, 169|23578|4, 1789|2356|4, 178|23569|4.
a(10) = 9: 1|2356|4|78(10)|9, 1|23578|4|6(10)|9, 1|258(10)|367|4|9, 1|258(10)|36|4|79, 1|259|36|4|78(10), 1|267(10)|358|4|9, 1|268|357(10)|4|9, 1|27|3589|4|6(10), 1|27|358|4|69(10).

Cf. A252897, A275780, A278329, A278340, A281994.

A361911 Number of set partitions of {1..n} with block-medians summing to an integer.

Original entry on oeis.org

1, 1, 3, 10, 30, 107, 479, 2249, 11173, 60144, 351086, 2171087, 14138253, 97097101, 701820663, 5303701310, 41838047938, 343716647215, 2935346815495, 25999729551523, 238473713427285, 2261375071834708, 22141326012712122, 223519686318676559, 2323959300370456901

Offset: 1

Gus Wiseman, Apr 14 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{124}{3}}
                                {{13}{24}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition {{1,4},{2,3}} has medians {5/2,5/2}, with sum 5, so is counted under a(4).

For median instead of sum we have A361864.
For mean of means we have A361865.
For mean instead of median we have A361866.
A000110 counts set partitions.
A000975 counts subsets with integer median, mean A327475.
A013580 appears to count subsets by median, A327481 by mean.
A308037 counts set partitions with integer average block-size.
A325347 = partitions w/ integer median, complement A307683, strict A359907.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A007837, A035470, A038041, A079309, A231147, A275714, A275780, A359893, A361801, A361910.

sps[{}]:={{}}; sps[set:{i_,_}] := Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], IntegerQ[Total[Median/@#]]&]],{n,10}]

a(12)-a(25) from Christian Sievers, Aug 26 2024

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 5, 9, 16, 27, 48, 78, 133

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A292432 at a(9) = 48, A292432(9) = 46.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
but none of these has distinct block-sums, so m is counted under a(5).
The a(2) = 1 through a(6) = 9 normal multisets:
  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
                  {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
                  {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
                             {1,1,2,2,2}  {1,1,1,1,2,3}
                             {1,2,2,2,2}  {1,1,1,2,2,2}
                                          {1,1,2,2,2,2}
                                          {1,2,2,2,2,2}
                                          {1,2,2,2,2,3}
                                          {1,2,3,3,3,3}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Without distinct sums we have A292432, complement A382214.
The strongly normal version without distinct sums is A292444, complement A381996.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
The complement is counted by A382216.
The strongly normal version is A382430, complement A382460.
The case of a unique choice is counted by A382459, without distinct sums A382458.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.

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[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]

A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

Original entry on oeis.org

1, 0, 1, 5, 61, 2721, 788221

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. 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(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 61 antichains:
  {1234}  {12}{34}    {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {13}{24}    {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {12}{134}   {12}{13}{34}   {12}{13}{23}{24}
          {12}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {13}{124}   {12}{23}{24}   {12}{13}{24}{34}
          {13}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {14}{123}   {12}{24}{34}   {12}{23}{24}{34}
          {14}{234}   {13}{14}{24}   {13}{14}{24}{34}
          {23}{124}   {13}{23}{24}   {13}{23}{24}{34}
          {23}{134}   {13}{23}{34}   {12}{13}{14}{234}
          {24}{134}   {13}{24}{34}   {12}{23}{24}{134}
          {34}{123}   {14}{24}{34}   {123}{124}{134}{234}
          {123}{124}  {12}{13}{234}
          {123}{134}  {12}{14}{234}
          {123}{234}  {12}{23}{134}
          {124}{134}  {12}{24}{134}
          {124}{234}  {13}{14}{234}
          {134}{234}  {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}

Antichain covers are A006126.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums and no singletons are A326565.
Antichain covers with different edge-sizes and no singletons are A326569.
The case with singletons allowed is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A035470, A307249, A321469, A326519, A326566, A326570, A326573.

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}]

A326573 Number of connected antichains of subsets of {1..n}, all having different sums.

Original entry on oeis.org

1, 1, 1, 5, 59, 2689, 787382

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. 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(3) = 5 antichains:
  {{1,2,3}}
  {{1,3},{2,3}}
  {{1,2},{2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 59 antichains:
  {1234}  {12}{134}   {12}{13}{14}   {12}{13}{14}{24}   {12}{13}{14}{24}{34}
          {12}{234}   {12}{13}{24}   {12}{13}{14}{34}   {12}{13}{23}{24}{34}
          {13}{124}   {12}{13}{34}   {12}{13}{23}{24}
          {13}{234}   {12}{14}{34}   {12}{13}{23}{34}
          {14}{123}   {12}{23}{24}   {12}{13}{24}{34}
          {14}{234}   {12}{23}{34}   {12}{14}{24}{34}
          {23}{124}   {12}{24}{34}   {12}{23}{24}{34}
          {23}{134}   {13}{14}{24}   {13}{14}{24}{34}
          {24}{134}   {13}{23}{24}   {13}{23}{24}{34}
          {34}{123}   {13}{23}{34}   {12}{13}{14}{234}
          {123}{124}  {13}{24}{34}   {12}{23}{24}{134}
          {123}{134}  {14}{24}{34}   {123}{124}{134}{234}
          {123}{234}  {12}{13}{234}
          {124}{134}  {12}{14}{234}
          {124}{234}  {12}{23}{134}
          {134}{234}  {12}{24}{134}
                      {13}{14}{234}
                      {13}{23}{124}
                      {14}{34}{123}
                      {23}{24}{134}
                      {12}{134}{234}
                      {13}{124}{234}
                      {14}{123}{234}
                      {23}{124}{134}
                      {123}{124}{134}
                      {123}{124}{234}
                      {123}{134}{234}
                      {124}{134}{234}

Antichain covers are A006126.
Connected antichains are A048143.
Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
Antichain covers with equal edge-sums are A326566.
The non-connected case is A326572.
Cf. A000372, A293510, A307249, A321469, A323818, A326519, A326565, A326569, A326570, A326571.

A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

Original entry on oeis.org

1, 0, 3, 2, 12, 18, 101, 232, 1547, 3768, 24974, 116728, 687419, 3489664, 26436217, 159031250, 1129056772

Offset: 1

Gus Wiseman, Apr 04 2023

			The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}  {{12345}}
            {{13}{2}}    {{123}{4}}  {{1245}{3}}
            {{1}{2}{3}}              {{135}{24}}
                                     {{15}{234}}
                                     {{1}{234}{5}}
                                     {{12}{3}{45}}
                                     {{135}{2}{4}}
                                     {{14}{25}{3}}
                                     {{15}{24}{3}}
                                     {{1}{24}{3}{5}}
                                     {{15}{2}{3}{4}}
                                     {{1}{2}{3}{4}{5}}

For median instead of mean we have A361864.
For sum instead of outer mean we have A361866, median A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions whose block-sizes have integer mean.
A327475 counts subsets with integer mean, median A000975.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]

a(13)-a(17) from Christian Sievers, Jun 30 2025

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1

Offset: 0

Gus Wiseman, Apr 16 2024

			The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
     1
     0     1
     0     1     1
     0     2     2     1
     0     2     8     4     1
     0     2    19    24     6     1
     0     2    47    95    49     9     1
     0     6   105   363   297    93    12     1
     0    12   248  1292  1660   753   158    16     1
     0    11   563  4649  8409  5591  1653   250    20     1
     0     2  1414 15976 41264 38074 15590  3249   380    25     1
Row n = 4 counts the following set partitions:
  .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
     {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
                    {{1,2},{3,4}}    {{1},{2,4},{3}}
                    {{1,3},{2},{4}}  {{1,4},{2},{3}}
                    {{1,3},{2,4}}
                    {{1,2,3},{4}}
                    {{1,2,4},{3}}
                    {{1,3,4},{2}}

Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 0, 7, 3, 11, 18, 9

Offset: 0

Gus Wiseman, Mar 30 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,2,2,2,2,3,3,4} has three multiset partitions into a set of sets:
  {{2},{1,2},{2,3},{2,3,4}}
  {{2},{2,3},{2,4},{1,2,3}}
  {{2},{3},{1,2},{2,3},{2,4}}
so is not counted under a(8).
The a(1) = 1 through a(7) = 7 normal multisets:
  {1}  .  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  .  {1,1,1,1,2,3,4}
          {1,2,2}             {1,2,2,2,3}     {1,1,1,2,2,2,3}
                              {1,2,3,3,3}     {1,1,1,2,3,3,3}
                                              {1,2,2,2,2,3,4}
                                              {1,2,2,2,3,3,3}
                                              {1,2,3,3,3,3,4}
                                              {1,2,3,4,4,4,4}

For constant instead of strict blocks we have A000045.
Factorizations of this type are counted by A050326, with distinct sums A381633.
For the strong case see A292444, A382430, complement A381996, A382523.
MM-numbers of sets of sets are A302494, see A302478, A382201.
Twice-partitions into distinct sets are counted by A358914, with distinct sums A279785.
For integer partitions we have A382079 (A293511), with distinct sums A382460, (A381870).
With distinct sums we have A382459.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions: A034691, A035310, A116539, A255906, A381718.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A292432, A317532, A326519, A382428.

allnorm[n_]:=If[n<=0,{{}},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[Select[allnorm[n], Length[Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]]==1&]], {n,0,5}]

A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 0, 2, 1, 3, 2, 7, 4, 10, 19

Offset: 0

Gus Wiseman, Apr 01 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,2,2,2,2,3,3,4} has only one multiset partition into a set of sets with distinct sums: {{2},{1,2},{2,3},{2,3,4}}, so is counted under a(8).
The a(1) = 1 through a(7) = 7 multisets:
  {1}  .  {112}  {1122}  {11123}  {111233}  {1111234}
          {122}          {12223}  {122233}  {1112223}
                         {12333}            {1112333}
                                            {1222234}
                                            {1222333}
                                            {1233334}
                                            {1234444}

Twice-partitions of this type are counted by A279785, A270995, A358914.
Factorizations of this type are counted by A381633, A050320, A050326.
Normal multiset partitions of this type are A381718, A116540, A116539.
Multiset partitions of this type are ranked by A382201, A302478, A302494.
For at least one choice: A382216 (strict A382214), complement A382202 (strict A292432).
For the strong case see: A382430 (strict A292444), complement A382523 (strict A381996).
Without distinct sums we have A382458.
For integer partitions we have A382460, ranks A381870, strict A382079, ranks A293511.
Set multipartitions: A089259, A296119, A318360.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Partitions: A382077 (A382200), A381992 (A382075), A382078 (A293243), A381990 (A381806).
Cf. A000110, A000670, A007716, A255903, A275780, A317532, A321469, A326519, A381078, A381441, A382428.

allnorm[n_]:=If[n<=0,{{}},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[Select[allnorm[n],Length[Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]]==1&]],{n,0,5}]

A326030 Number of antichains of subsets of {1..n} with different edge-sums.

Original entry on oeis.org

2, 3, 6, 19, 132, 3578, 826949

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(3) = 19 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{2}}      {{2}}
               {{1,2}}    {{3}}
               {{1},{2}}  {{1,2}}
                          {{1,3}}
                          {{2,3}}
                          {{1},{2}}
                          {{1,2,3}}
                          {{1},{3}}
                          {{2},{3}}
                          {{1},{2,3}}
                          {{2},{1,3}}
                          {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Set partitions with different block-sums are A275780.
MM-numbers of multiset partitions with different part-sums are A326535.
The covering case is A326572.
Antichains with equal edge-sums are A326574.
Cf. A000372, A003182, A006126, A321469, A326519, A326571, A326573.

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}]

cp's OEIS Frontend

A278339 Number of set partitions of [n] into subsets whose element sums are distinct squares.

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A361911 Number of set partitions of {1..n} with block-medians summing to an integer.

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A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

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A326571 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having different sums.

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A326573 Number of connected antichains of subsets of {1..n}, all having different sums.

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A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

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A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

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A382458 Number of normal multisets of size n that can be partitioned into a set of sets in exactly one way.

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A382459 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums in exactly one way.

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A326030 Number of antichains of subsets of {1..n} with different edge-sums.

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