A050326 -id:A050326 - cp's OEIS frontend

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

1, 3, 5, 16, 17, 140, 65, 1395, 2969, 22176, 1025, 1050766, 4097, 13010328, 128268897, 637598438, 65537, 64864962683, 262145, 1676258452736, 28683380484257, 24908619669860, 4194305, 30567710172480050, 8756434134071649, 62128557507554504, 21271147396968151093

Offset: 1

Andrew Howroyd, Jan 23 2020

nonn

The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
From Gus Wiseman, Apr 03 2025: (Start)
Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
         {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
                    {{1},{2},{2}}  {{1,2},{2,3}}
                    {{1},{2},{3}}  {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{1,3},{2,4}}
                                   {{1,4},{2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{2},{3}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A330942, A331639.
For constant instead of strict blocks we have A034729.
Without equal sizes we have A116540 (normal set multipartitions).
Without strict blocks we have A317583.
For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
For equal sums instead of sizes we have A382429, non-strict blocks A326518.
Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.

a(n) = Sum_{d|n} A330942(n/d, d).

a(p) = 2^(p-1) + 1 for prime p.

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}]

A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 34, 45

Offset: 0

Gus Wiseman, Apr 01 2025

First differs from A381996 at a(12) = 45, A381996(12) = 47.

We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.

			First differs from A381996 in not counting the following under a(12):
  {1,1,1,1,1,1,2,2,3,3,4,5}
  {1,1,1,1,2,2,2,2,3,3,3,3}
The a(1) = 1 through a(6) = 6 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}  {1,1,1,2,2,3}
              {1,2,3}  {1,1,2,3}  {1,1,2,2,3}  {1,1,1,2,3,4}
                       {1,2,3,4}  {1,1,2,3,4}  {1,1,2,2,3,3}
                                  {1,2,3,4,5}  {1,1,2,2,3,4}
                                               {1,1,2,3,4,5}
                                               {1,2,3,4,5,6}

Twice-partitions of this type are counted by A279785, strict A358914.
Factorizations of this type are counted by A381633, strict A050326.
Normal multiset partitions of this type are counted by A381718, strict A116539.
For integer partitions we have A381992, ranks A382075, complement A381990, ranks A381806.
The strict version is A381996.
The strict version for integer partitions is A382077, ranks A382200, complement A382078, ranks A293243.
The labeled version is A382216, complement A382202, strict A382214, complement A292432.
The complement is counted by A382430, strict A292444.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
Table[Length[Select[strnorm[n],Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A103774 Number of ways to write n! as product of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 2, 6, 10, 42, 42, 82, 204, 1196, 1556, 10324, 34668, 104948, 104964, 873540, 1309396, 11855027, 25238220, 91193575, 453628255, 5002616219, 5902762219, 21142729523, 122981607092, 189706055368, 547296181656, 7291700021313, 14330422534833, 202498591157970

Offset: 1

Reinhard Zumkeller, Feb 15 2005

nonn

a(n) = A050320(A000142(n)).
From Gus Wiseman, Aug 20 2020: (Start)
Also the number of set multipartitions (multisets of sets) of the multiset of prime factors of n!. For example, The a(2) = 1 through a(6) = 10 set multipartitions are:
  {1}  {12}    {1}{1}{12}    {1}{1}{123}      {1}{1}{12}{123}
       {1}{2}  {1}{1}{1}{2}  {1}{12}{13}      {1}{12}{12}{13}
                             {1}{1}{1}{23}    {1}{1}{1}{12}{23}
                             {1}{1}{2}{13}    {1}{1}{1}{2}{123}
                             {1}{1}{3}{12}    {1}{1}{2}{12}{13}
                             {1}{1}{1}{2}{3}  {1}{1}{3}{12}{12}
                                              {1}{1}{1}{1}{2}{23}
                                              {1}{1}{1}{2}{2}{13}
                                              {1}{1}{1}{2}{3}{12}
                                              {1}{1}{1}{1}{2}{2}{3}
(End)

			n=5, 5! = 1*2*3*4*5 = 120 = 2 * 2 * 2 * 3 * 5: a(5)=#{2*2*2*3*5,2*2*2*15,2*2*6*5,2*2*30,2*2*3*10,2*6*10}=6.

Index entries for sequences related to factorial numbers.

A103775 is the strict case.
A157612 is the case of superprimorials.
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.
A076716 counts factorizations of factorials.
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A157612 counts strict factorizations of factorials.
Cf. A000110, A005117, A008480, A124010, A318360.
Factorial numbers: A000142, A007489, A022559, A027423, A071626, A325272, A325617, A336498.

sub[w_, e_] := Block[{v=w}, v[[e]]--; v]; ric[w_, k_] := ric[w, k] = If[Max[w] == 0, 1, Block[{e, s, p = Flatten@ Position[Sign@w, 1]}, s = Select[ Prepend[#, First@p] & /@ Subsets[Rest@p], Total[1/2^#] <= k &]; Sum[ric[sub[w, e], Total[1/2^e]], {e, s}]]]; a[n_] := ric[ Sort[ Last /@ FactorInteger[n!]], 1]; Array[a, 22] (* Giovanni Resta, Sep 30 2019 *)

a(17)-a(18) from Amiram Eldar, Sep 30 2019

a(19)-a(31) from Giovanni Resta, Sep 30 2019

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

Original entry on oeis.org

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)).

A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

Original entry on oeis.org

1, 1, 2, 10, 141, 6769, 1298995, 1148840085, 5307091649182, 143026276277298216, 24801104674619158730662, 30190572492693121799801655311, 278937095127086600900558327826721594

Offset: 0

Gus Wiseman, Aug 15 2020

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

Also the number of set multipartitions (multisets 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*2*3  6*6*10
              2*5*6*6
              2*2*3*30
              2*2*6*15
              2*3*6*10
              2*2*3*5*6
              2*2*2*3*15
              2*2*3*3*10
              2*2*2*3*3*5
The a(1) = 1 through a(3) = 10 set multipartitions:
     {1}  {1}{12}    {1}{12}{123}
          {1}{1}{2}  {12}{12}{13}
                     {1}{1}{12}{23}
                     {1}{1}{2}{123}
                     {1}{2}{12}{13}
                     {1}{3}{12}{12}
                     {1}{1}{1}{2}{23}
                     {1}{1}{2}{2}{13}
                     {1}{1}{2}{3}{12}
                     {1}{1}{1}{2}{2}{3}

A000142 counts divisors of superprimorials.
A022915 counts permutations of the same multiset.
A103774 is the version for factorials instead of superprimorials.
A337073 is the strict case (strict factorizations into squarefree numbers).
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.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A089259 counts set multipartitions of integer partitions.
A116540 counts normal set multipartitions.
A317829 counts factorizations of superprimorials.
A337069 counts strict factorizations of superprimorials.
Cf. A000178, A002110, A027423, A124010, A181818, A303279, A318360, A336417, A337070.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facsqf[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsqf[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[facsqf[chern[n]]],{n,0,3}]

\\ See A318360 for count.
a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020

a(n) = A050320(A006939(n)).

a(n) = A318360(A002110(n)). - Andrew Howroyd, Aug 31 2020

a(7)-a(12) from Andrew Howroyd, Aug 31 2020

A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 4 strict set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2},{3},{1,2}}

Cf. A007716, A045778, A049311, A050326, A056239, A112798, A116539, A116540, A292432, A292444, A293511, A318284, A318360, A318369.

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

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

cp's OEIS Frontend

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

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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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A382523 Number of non-isomorphic finite multisets of size n that can be partitioned into sets with distinct sums.

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A103774 Number of ways to write n! as product of squarefree numbers.

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

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A337072 Number of factorizations of the superprimorial A006939(n) into squarefree numbers > 1.

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A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

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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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