A318361 -id:A318361 - cp's OEIS frontend

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

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}]

A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

Original entry on oeis.org

1, 2, 6, 20, 74, 311, 1401, 6913, 36376, 205421, 1228288, 7786802, 51937607, 364250763, 2673314121, 20504809133, 163844631872, 1361874185139, 11748149246269, 105029750531640, 971403871953460, 9282643841237360, 91519776792040324, 929892817423282068, 9725646244888190337

Offset: 1

Gus Wiseman, Aug 25 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(4) = 20 sets of sets:
  {{1,2,3,4}}
  {{1},{1,2,3}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{3,4}}
  {{1},{3},{1,2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

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

Cf. A007716, A049311, A050326, A116540, A283877, A292432, A292444, A318361, A318369, A318370.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A321176 Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 7, 10, 15, 21, 28

Offset: 0

Gus Wiseman, Oct 29 2018

A set system is a finite set of finite nonempty sets.

			The a(2) = 1 through a(9) = 15 partitions:
  (11)  (111)  (211)   (221)    (222)     (322)      (2222)      (333)
               (1111)  (2111)   (2211)    (2221)     (3221)      (3222)
                       (11111)  (3111)    (3211)     (3311)      (3321)
                                (21111)   (22111)    (22211)     (4221)
                                (111111)  (31111)    (32111)     (22221)
                                          (211111)   (41111)     (32211)
                                          (1111111)  (221111)    (33111)
                                                     (311111)    (42111)
                                                     (2111111)   (222111)
                                                     (11111111)  (321111)
                                                                 (411111)
                                                                 (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 10 integer partitions together with a realizing set system for each (the parts of the partition count the appearances of each vertex in the set system):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3311): {{1,2},{1,2,3},{1,2,4}}
      (3221): {{1,2},{1,3},{1,2,3,4}}
     (32111): {{1,2},{1,3},{1,2,4,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{3,4},{1,2,3,4}}
     (22211): {{1,2,3},{1,2,3,4,5}}
    (221111): {{1,2},{1,2,3,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Cf. A000070, A000569, A147878, A209816, A283877, A306005, A318361, A320922, A320923, A320924, A321177.

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]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],hyp[#]!={}&]],{n,8}]

A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 1, 2, 14, 422, 59433, 43181280, 178025660042, 4550598470020490, 782250333882971717562, 974196106965358319940100513, 9412280190038329162111356578977100, 751537739224674099813783040471383322758327

Offset: 0

Gus Wiseman, Aug 15 2020

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). It has n! divisors.

Also the number of strict set multipartitions (sets of sets) of the multiset of prime factors of the superprimorial A006939(n).

			The a(1) = 1 through a(3) = 10 factorizations:
    2  2*6  2*6*30    2*6*30*210
            2*3*6*10  6*10*30*42
                      2*3*6*30*70
                      2*5*6*30*42
                      2*3*10*30*42
                      2*3*6*10*210
                      2*6*10*15*42
                      2*6*10*21*30
                      2*6*14*15*30
                      3*6*10*14*30
                      2*3*5*6*10*42
                      2*3*5*6*14*30
                      2*3*6*7*10*30
                      2*3*6*10*14*15
The a(1) = 1 through a(3) = 14 set multipartitions:
    {1}  {1}{12}  {1}{12}{123}    {1}{12}{123}{1234}
                  {1}{2}{12}{13}  {12}{13}{123}{124}
                                  {1}{12}{13}{23}{124}
                                  {1}{12}{13}{24}{123}
                                  {1}{12}{14}{23}{123}
                                  {1}{2}{12}{123}{134}
                                  {1}{2}{12}{13}{1234}
                                  {1}{2}{13}{123}{124}
                                  {1}{3}{12}{123}{124}
                                  {2}{12}{13}{14}{123}
                                  {1}{2}{12}{13}{14}{23}
                                  {1}{2}{12}{4}{13}{123}
                                  {1}{2}{3}{12}{13}{124}
                                  {1}{2}{3}{12}{14}{123}

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103775 is the version for factorials instead of superprimorials.
A337072 is the non-strict version.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A045778 counts strict factorizations.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts strict set multipartitions of integer partitions.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A283877 counts non-isomorphic strict set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A022559, A103774, A124010, A181818, A318361, A336417, A337070.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
ystfac[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[ystfac[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[ystfac[chern[n]]],{n,0,4}]

\\ See A318361 for count.
a(n) = {if(n==0, 1, count(vector(n, i, i)))} \\ Andrew Howroyd, Sep 01 2020

a(n) = A050326(A006939(n)).

a(n) = A318361(A002110(n)). - Andrew Howroyd, Sep 01 2020

a(7)-a(13) from Andrew Howroyd, Sep 01 2020

A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 1, 2, 4, 12, 38, 145, 586, 2619, 12096, 58370, 285244, 1436815, 7281062, 37489525, 193417612

Offset: 0

Gus Wiseman, Mar 14 2025

			The a(1) = 1 through a(5) = 12 multisets:
  {1}  {1,2}  {1,2,3}    {1,2,3,4}      {1,2,3,4,5}
              {1,1,2,2}  {1,1,2,2,4}    {1,1,2,2,4,5}
                         {1,1,2,3,3}    {1,1,2,3,3,5}
                         {1,1,1,2,2,3}  {1,1,2,3,4,4}
                                        {1,2,2,3,3,4}
                                        {1,1,1,2,2,3,5}
                                        {1,1,1,2,2,4,4}
                                        {1,1,1,2,3,3,4}
                                        {1,1,2,2,2,3,4}
                                        {1,1,2,2,3,3,3}
                                        {1,1,1,1,2,2,3,4}
                                        {1,1,1,2,2,2,3,3}

Set systems: A050342, A116539, A296120, A318361.
The number of possible choices was A152827, non-strict A058694.
Set multipartitions with distinct sums: A279785, A381718.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Constant instead of strict partitions: A381807, A066843.
A000041 counts integer partitions, strict A000009, constant A000005.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@Range[n]]]],{n,0,10}]

a(12)-a(16) from Christian Sievers, Jun 04 2025

A321177 Heinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.

Original entry on oeis.org

1, 4, 8, 12, 16, 18, 24, 27, 32, 36, 40

Offset: 1

Gus Wiseman, Oct 29 2018

A set system is a finite set of finite nonempty sets.

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

			Each term paired with its Heinz partition and a realizing set system:
  1:       (): {}
  4:     (11): {{1,2}}
  8:    (111): {{1,2,3}}
  12:   (211): {{1,2},{1,3}}
  16:  (1111): {{1,2,3,4}}
  18:   (221): {{1,2},{1,2,3}}
  24:  (2111): {{1,2},{1,3,4}}
  27:   (222): {{1,2},{1,3},{2,3}}
  32: (11111): {{1,2,3,4,5}}
  36:  (2211): {{1,2},{1,2,3,4}}
  40:  (3111): {{1,2},{1,3},{1,4}}

Cf. A000070, A000569, A056239, A112798, A283877, A306005, A318361, A320922, A320923, A320924, A320925, A321176.

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]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[20],!hyp[nrmptn[#]]=={}&]

A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 9, 11, 17, 20

Offset: 0

Gus Wiseman, Oct 29 2018

A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (211)   (2111)   (222)     (2221)     (2222)      (3222)
               (1111)  (11111)  (2211)    (22111)    (3221)      (22221)
                                (3111)    (31111)    (22211)     (32211)
                                (21111)   (211111)   (32111)     (33111)
                                (111111)  (1111111)  (41111)     (222111)
                                                     (221111)    (321111)
                                                     (311111)    (411111)
                                                     (2111111)   (2211111)
                                                     (11111111)  (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 9 integer partitions together with a realizing strict antichain for each (the parts of the partition count the appearances of each vertex in the antichain):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3221): {{1,2},{1,3},{1,4},{2,3}}
     (32111): {{1,3},{1,2,4},{1,2,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{1,3,4},{2,3,4}}
     (22211): {{1,2,3,4},{1,2,3,5}}
    (221111): {{1,2,3},{1,2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
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]]]];
anti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],anti[#]!={}&]],{n,8}]

A321188 Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 11, 0, 0, 0, 4, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Oct 29 2018

A set system is a finite set of finite nonempty sets.

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(36) = 4 set systems with no singletons whose multiset union is {1,1,2,2,3,4}:
  {{1,2},{1,2,3,4}}
  {{1,2,3},{1,2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}

Cf. A000070, A000296, A000569, A050326, A056239, A112798, A283877, A292444, A305936, A306005, A318285, A318361, A320922, A320923, A320924.

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]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[hyp[nrmptn[n]]],{n,30}]

A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 1, 0, 11, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 13, 1, 1, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 0, 1, 41, 0, 0, 0, 0, 0, 1, 0, 20, 0, 0, 2, 0, 0, 0, 0, 6, 16, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Nov 25 2018

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(90) = 7 set multipartitions of {1,1,1,2,2,3,3,4} with no singletons:
  {{1,2},{1,2},{1,3},{3,4}}
  {{1,2},{1,3},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{1,2,3,4}}
  {{1,2},{1,2,3},{1,3,4}}
  {{1,3},{1,2,3},{1,2,4}}
  {{1,4},{1,2,3},{1,2,3}}

Cf. A001055, A007716, A049311, A056156, A116540, A181821, A255906, A304382.

Cf. A318284, A318286, A318360, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]];
Table[Length[sqnopfacs[Times@@Prime/@nrmptn[n]]],{n,30}]

cp's OEIS Frontend

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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A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

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A321176 Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.

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A337073 Number of strict factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

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A381808 Number of multisets that can be obtained by choosing a strict integer partition of m for each m = 0..n and taking the multiset union.

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A321177 Heinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.

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A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

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A321188 Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n).

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A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

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