A050342 -id:A050342 - cp's OEIS frontend

A103775 Number of ways to write n! as product of distinct squarefree numbers.

1, 1, 2, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Feb 15 2005

nonn

From Gus Wiseman, Aug 23 2020: (Start)
Also the number of set-systems (sets of sets) whose multiset union is the multiset of prime factors of n!. For example, the a(1) = 1 through a(7) = 3 set-systems (empty columns indicated by dots) are:
  0  {1}  {1,2}    .  {1},{1,2},{1,3}  .  {1},{1,2},{1,3},{1,2,4}
          {1},{2}                         {1},{1,2},{1,4},{1,2,3}
                                          {1},{2},{1,2},{1,3},{1,4}
(End)

			n=7, 7! = 1*2*3*4*5*6*7 = 5040 = 2*2*2*2*3*3*5*7: a(7) = #{2*3*6*10*14, 2*6*10*42, 2*6*14*30} = 3.

Index entries for sequences related to factorial numbers.

A103774 is the non-strict version.
A337073 is the version for superprimorials, with non-strict version A337072.
A001055 counts factorizations.
A045778 counts strict factorizations.
A048656 counts squarefree divisors of factorials.
A050320 counts factorizations into squarefree numbers.
A050326 counts strict factorizations into squarefree numbers.
A050342 counts set-systems by total sum.
A076716 counts factorizations of factorials.
A116539 counts set-systems covering an initial interval.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A089259, A116540, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

yst[n_]:=yst[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[yst[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[yst[n!]],{n,15}] (* Gus Wiseman, Aug 21 2020 *)

a(n) = 0 for n > 7;

a(n) = A050326(A000142(n)).

A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

Original entry on oeis.org

1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855

Offset: 1

Gus Wiseman, Jan 31 2018

nonn

			The a(5) = 7 compositions:
      (5) = (5)
     (41) = (4)*(1)
     (14) = (14)
     (32) = (3)*(2)
     (23) = (23)
    (131) = (13)*(1)
    (212) = (2)*(12)
Not included:
    (311) = (3)*(1)*(1)
    (113) = (113)
    (221) = (2)*(2)*(1)
    (122) = (122)
   (2111) = (2)*(1)*(1)*(1)
   (1211) = (12)*(1)*(1)
   (1121) = (112)*(1)
   (1112) = (1112)
  (11111) = (1)*(1)*(1)*(1)*(1)

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877, A292432, A296373, A299023, A299026, A299027.

nn=50;
ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018

Weigh transform of A032153.

A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 4, 6, 10, 18, 30, 50, 86, 145, 245, 417, 705, 1193, 2024, 3427, 5804, 9836, 16660, 28220, 47811, 80991, 137197, 232423, 393729, 666982, 1129898, 1914078, 3242495, 5492898, 9305130, 15763154, 26703273, 45236138, 76631348, 129815818, 219911870, 372537244, 631089250

Offset: 0

Ilya Gutkovskiy, Jul 30 2018

nonn

Invert transform of A111133.

Alois P. Heinz, Table of n, a(n) for n = 0..4371
N. J. A. Sloane, Transforms

Cf. A000009, A050342, A089259, A111133, A270995, A279785, A299106, A304969, A305651, A317535.

seq(coeff(series(1/(1+1/(1-x)-mul(1+x^k,k=1..n)), x,n+1),x,n),n=0..50); # Muniru A Asiru, Jul 30 2018
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*(b(i)-1), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 16 2022

nmax = 42; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[(PartitionsQ[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsQ[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

G.f.: 1/(1 - Sum_{k>=1} A111133(k)*x^k).

A360764 Number T(n,k) of sets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=max(i:T(n,i)>0), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 1, 4, 6, 1, 0, 1, 6, 8, 4, 0, 1, 6, 13, 9, 1, 0, 1, 8, 18, 16, 6, 0, 1, 8, 24, 29, 13, 2, 0, 1, 10, 30, 43, 29, 6, 0, 1, 10, 39, 64, 52, 19, 1, 0, 1, 12, 46, 89, 89, 42, 7, 0, 1, 12, 56, 122, 139, 85, 22, 1

Offset: 0

Alois P. Heinz, Feb 19 2023

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

			T(6,1) = 1: {[6]}.
T(6,2) = 4: {[1],[5]}, {[2],[4]}, {[1,5]}, {[2,4]}.
T(6,3) = 6: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[2],[3]}.
T(6,4) = 1: {[1],[2],[1,2]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1,  2;
  0, 1,  2,  1;
  0, 1,  4,  2;
  0, 1,  4,  6,  1;
  0, 1,  6,  8,  4;
  0, 1,  6, 13,  9,  1;
  0, 1,  8, 18, 16,  6;
  0, 1,  8, 24, 29, 13,  2;
  0, 1, 10, 30, 43, 29,  6;
  0, 1, 10, 39, 64, 52, 19, 1;
  ...

Alois P. Heinz, Rows n = 0..250, flattened

Columns k=0-2 give: A000007, A057427, A052928(n-1) for n>=3.
Row sums give A050342.
Cf. A000009, A008289, A055884, A330462, A360742, A360763.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i<0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]] ;
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

A050344 Number of partitions of n into distinct parts with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 11, 25, 60, 141, 321, 742, 1688, 3810, 8580, 19225, 42844, 95156, 210480, 463866, 1018957, 2231114, 4870400, 10601805, 23015117, 49833471, 107636878, 231940988, 498671281, 1069826434, 2290402343, 4893782240, 10436263572, 22214850439, 47202869437

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

			4 = (((4))) = (((3)))+(((1))) = (((3))+((1))) = ((3)+(1)) = ((3+1)) = ((2+1))+((1)) = ((2+1)+(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A050342-A050350.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*h(n-i*j, i-1), j=0..n/i)))
    end:
f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i, i), j)*f(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(f(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]];
h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i, i], j]* h[n - i*j, i - 1], {j, 0, n/i}]]];
f[n_, i_] := f[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[h[i, i], j]* f[n - i*j, i - 1], {j, 0, n/i}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[f[i, i], j]* b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Weigh transform of A050343.

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

A358913 Number of finite sequences of distinct sets with total sum n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 28, 45, 86, 172, 344, 608, 1135, 2206, 4006, 7689, 13748, 25502, 47406, 86838, 157560, 286642, 522089, 941356, 1718622, 3079218, 5525805, 9902996, 17788396, 31742616, 56694704, 100720516, 178468026, 317019140, 560079704, 991061957

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(5) = 11 sequences of sets:
  ({1})  ({2})  ({3})      ({4})        ({5})
                ({1,2})    ({1,3})      ({1,4})
                ({1},{2})  ({1},{3})    ({2,3})
                ({2},{1})  ({3},{1})    ({1},{4})
                           ({1},{1,2})  ({2},{3})
                           ({1,2},{1})  ({3},{2})
                                        ({4},{1})
                                        ({1},{1,3})
                                        ({1,2},{2})
                                        ({1,3},{1})
                                        ({2},{1,2})

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The unordered version is A050342, non-strict A261049.
The case of strictly decreasing sums is A279785.
This is the distinct case of A304969.
The case of distinct sums is A336343, constant sums A279791.
This is the case of A358906 with strict partitions.
The version for compositions instead of strict partitions is A358907.
The case of twice-partitions is A358914.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A063834 counts twice-partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf. A000009, A000041, A000219, A271619, A296122, A336342, A358908.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 13 2024

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]

a(n) = Sum_{k} A330462(n,k) * k!.

A305651 Expansion of Product_{k>=1} (1 + x^k)^(q(k)-1), where q(k) = number of partitions of k into distinct parts (A000009).

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 26, 39, 59, 87, 132, 192, 284, 419, 612, 892, 1303, 1887, 2730, 3945, 5677, 8154, 11689, 16711, 23839, 33960, 48244, 68432, 96888, 136922, 193148, 272058, 382508, 537007, 752735, 1053550, 1472406, 2054988, 2863993, 3986245, 5541008

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

Weigh transform of A111133.

Convolution of the sequences A050342 and A081362.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A000009, A050342, A081362, A089259, A111133, A304966, A304969.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Jun 07 2018

nmax = 45; CoefficientList[Series[Product[(1 + x^k)^(PartitionsQ[k] - 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[(-1)^(k + 1)/k (1/ QPochhammer[x^k, x^(2 k)] - 1/(1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (PartitionsQ[d] - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} (1 + x^k)^A111133(k).

G.f.: Product_{k>=1} (1 + x^k)^(A000009(k)-1).

A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(10) = 13 clutters:
  {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
                {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
                                  {{2,3}}  {{2,4}}    {{2,5}}
                                           {{1,2,3}}  {{3,4}}
                                                      {{1,2,4}}
                                                      {{1,2},{1,3}}
.
  {{8}}          {{9}}          {{10}}
  {{1,7}}        {{1,8}}        {{1,9}}
  {{2,6}}        {{2,7}}        {{2,8}}
  {{3,5}}        {{3,6}}        {{3,7}}
  {{1,2,5}}      {{4,5}}        {{4,6}}
  {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
  {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
  {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
                 {{1,2},{1,5}}  {{2,3,5}}
                 {{1,2},{2,4}}  {{1,2,3,4}}
                 {{1,3},{1,4}}  {{1,2},{1,6}}
                 {{1,3},{2,3}}  {{1,2},{2,5}}
                                {{1,3},{1,5}}

Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.

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]]]];
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]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

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

cp's OEIS Frontend

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