A261049 -id:A261049 - cp's OEIS frontend

A320328 Number of square multiset partitions of integer partitions of n.

1, 1, 2, 3, 6, 11, 20, 36, 65, 117, 214, 382, 679

Offset: 0

Gus Wiseman, Oct 11 2018

A multiset partition is square if its length is equal to its number of distinct atoms.

			The a(1) = 1 through a(6) = 20 square partitions:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,1,1}}  {{2,2}}      {{1},{4}}      {{3,3}}
                  {{1},{2}}  {{1},{3}}    {{2},{3}}      {{1},{5}}
                             {{1,1,1,1}}  {{1},{1,3}}    {{2,2,2}}
                             {{1},{1,2}}  {{1},{2,2}}    {{2},{4}}
                             {{2},{1,1}}  {{2},{1,2}}    {{1},{1,4}}
                                          {{3},{1,1}}    {{4},{1,1}}
                                          {{1,1,1,1,1}}  {{1},{1,1,3}}
                                          {{1},{1,1,2}}  {{1,1},{1,3}}
                                          {{1,1},{1,2}}  {{1},{1,2,2}}
                                          {{2},{1,1,1}}  {{1,1},{2,2}}
                                                         {{1,2},{1,2}}
                                                         {{1},{2},{3}}
                                                         {{2},{1,1,2}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1},{1,1,1,2}}
                                                         {{1,1},{1,1,2}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Cf. A000219, A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A319616, A320330, A320331.

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[Join@@mps/@IntegerPartitions[n],Length[#]==Length[Union@@#]&]],{n,8}]

A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

Original entry on oeis.org

1, 1, 2, 6, 10, 23, 50, 95, 188, 378, 747, 1414, 2739, 5179, 9811, 18562, 34491, 64131, 118607, 218369, 400196, 731414, 1328069, 2406363, 4346152, 7819549, 14027500, 25090582, 44749372, 79586074, 141214698, 249882141, 441176493, 777107137, 1365801088, 2395427040, 4192702241

Offset: 0

Gus Wiseman, Dec 09 2022

nonn

			The a(1) = 1 through a(4) = 10 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((11)(1))  ((1)(3))
                            ((3)(1))
                            ((11)(2))
                            ((21)(1))
                            ((111)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..200

This is the distinct case of A055887 with weakly decreasing lengths.
This is the distinct case is A141199.
The case of distinct lengths also is A358836.
This is the case of A358906 with weakly decreasing lengths.
A000041 counts integer partitions, strict A000009.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all distinct Omegas.
A358912 counts sequences of partitions with distinct lengths.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000219, A261049, A271619, A296122, A358831, A358901, A358905, A358907.

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

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,v) = {[subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, #v, (1 + y*x^k + O(x*x^n))^v[k] ))]}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, Ser(R(n, Vec(polcoef(g, k, y), -n)))  ))} \\ Andrew Howroyd, Dec 31 2022

Terms a(16) and beyond from Andrew Howroyd, Dec 31 2022

A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 18, 3, 3, 3, 23, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 45, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 34, 3, 12, 1, 8, 3, 12, 1, 66, 1, 3, 8, 8, 3, 12, 1, 45, 8, 3

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

			The a(12) = 8 twice-factorizations are (2)*(2*3), (2)*(6), (3)*(2*2), (3)*(4), (2*2*3), (2*6), (3*4), (12).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000009, A005117, A045778, A261049, A271619, A281113, A294788, A295923, A296119, A296121.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Times@@(Length[facs[#]]&/@f),{f,Select[facs[n],UnsameQ@@#&]}],{n,100}]

A001055(n, m=n) = if(1==n, 1, sumdiv(n, d, if((d>1)&&(d<=m), A001055(n/d, d))));
A296118(n, m=n) = ((n<=m)*A001055(n) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA001055(d)*A296118(n/d, d-1)))); \\ Antti Karttunen, Oct 08 2018

Dirichlet g.f.: Product_{n > 1}(1 + A001055(n)/n^s).

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Gus Wiseman, Aug 18 2025

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 axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).
Allowing repeated partitions gives A299200, A357977, A357982, A357978.
For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.
For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.
For prime factors instead of partitions see A355741, A355742, A387136.
The disjoint case is A383706.
For initial intervals instead of partitions we have A387111.
The case of strict partitions is A387115.
The case of constant partitions is A387120.
Taking each prime factor (instead of index) gives A387133.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A008284, A063834, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]

A358906 Number of finite sequences of distinct integer partitions with total sum n.

Original entry on oeis.org

1, 1, 2, 7, 13, 35, 87, 191, 470, 1080, 2532, 5778, 13569, 30715, 69583, 160386, 360709, 814597, 1824055, 4102430, 9158405, 20378692, 45215496, 100055269, 221388993, 486872610, 1069846372, 2343798452, 5127889666, 11186214519, 24351106180, 52896439646

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 13 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((211))
                 ((2)(1))   ((1111))
                 ((1)(11))  ((1)(3))
                 ((11)(1))  ((3)(1))
                            ((11)(2))
                            ((1)(21))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

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

This is the case of A055887 with distinct partitions.
The unordered version is A261049.
The case of twice-partitions is A296122.
The case of distinct sums is A336342, constant sums A279787.
The version for sequences of compositions is A358907.
The case of weakly decreasing lengths is A358908.
The case of distinct lengths is A358912.
The version for strict partitions is A358913, distinct case of A304969.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all distinct Omegas.
Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      binomial(combinat[numbpart](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..32);  # Alois P. Heinz, Feb 13 2024

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

a(n) = Sum_{k} A330463(n,k) * k!.

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

Original entry on oeis.org

1, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 4, 0, 2, 2, 4, 0, 6, 2, 3, 0, 2, 4, 0, 0, 4, 4, 2, 0, 4, 2, 6, 0, 6, 4, 8, 0, 2, 6, 4, 0, 4, 3, 4, 0, 6, 2, 4, 0, 5, 0, 4, 0, 8, 4, 2, 0, 6, 2, 6, 0, 8, 4, 2, 0, 6, 6, 6, 0, 4, 6, 4, 0, 6, 8, 4, 0, 0, 2, 2, 0, 4, 4, 8

Offset: 1

Gus Wiseman, Aug 26 2025

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 prime indices of 90 are {1,2,2,3}, with choices:
  ((1),(2),(1,1),(3))
  ((1),(1,1),(2),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(1,1,1))
so a(90) = 4.

For multiset systems see A355529, set systems A367901.
For not necessarily different choices we have A355731, see A355740.
For divisors instead of constant partitions we have A355739 (also the disjoint case).
For prime factors instead of constant partitions we have A387136.
For all instead of just constant partitions we have A387110, disjoint case A383706.
For initial intervals instead of partitions we have A387111.
For strict instead of constant partitions we have A387115.
Twice partitions of this type are counted by A387179, constant-block case of A296122.
Positions of zero are A387180 (non-choosable), complement A387181 (choosable).
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]],UnsameQ@@#&]],{n,100}]

A320330 Number of T_0 multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 50, 100, 195, 366, 707, 1333, 2440

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 25 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,1,1}}      {{2,2}}            {{1,1,3}}
         {{1},{1}}  {{1},{2}}      {{1,1,2}}          {{1,2,2}}
                    {{1},{1,1}}    {{1},{3}}          {{1},{4}}
                    {{1},{1},{1}}  {{2},{2}}          {{2},{3}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{2},{1,1}}        {{1},{2,2}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{3},{1,1}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{2},{1,1}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, A320331.

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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@dual[#]&]],{n,8}]

A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.

Original entry on oeis.org

1, 1, 2, 5, 16, 54, 212, 834, 3558, 15394, 69512, 313107, 1474095, 6877031, 32877196

Offset: 0

Gus Wiseman, Nov 11 2018

a(n) is the number of integer partitions finer than (n, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + n ordered by refinement.

a(n+1)/a(n) appears to converge as n -> oo. - Chai Wah Wu, Nov 14 2018

			The a(1) = 1 through a(4) = 16 partitions:
  (1)  (21)   (321)     (4321)
       (111)  (2211)    (32221)
              (3111)    (33211)
              (21111)   (42211)
              (111111)  (43111)
                        (222211)
                        (322111)
                        (331111)
                        (421111)
                        (2221111)
                        (3211111)
                        (4111111)
                        (22111111)
                        (31111111)
                        (211111111)
                        (1111111111)
The partition (222211) is the combination of (22)(21)(2)(1), so is counted under a(4). The partition (322111) is the combination of (22)(3)(11)(1), (31)(21)(2)(1), or (211)(3)(2)(1), so is also counted under a(4).

Cf. A000217, A001970, A002846, A063834, A066723, A173519, A213427, A242422, A261049, A265947, A271619, A299201, A300383, A317141.

Cf. A321467, A321468, A321471, A321472, A321514.

Table[Length[Union[Sort/@Join@@@Tuples[IntegerPartitions/@Range[1,n]]]],{n,6}]

from collections import Counter
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A321470_gen(): # generator of terms
    aset = {(1,)}
    yield 1
    for n in count(2):
        yield len(aset)
        aset = {tuple(sorted(p+q)) for p in aset for q in (tuple(sorted(Counter(q).elements())) for q in partitions(n))}
A321470_list = list(islice(A321470_gen(),10)) # Chai Wah Wu, Sep 20 2023

a(n) <= A173519(n). - David A. Corneth, Sep 20 2023

a(9)-a(11) from Alois P. Heinz, Nov 12 2018
a(12)-a(13) from Chai Wah Wu, Nov 13 2018
a(14) from Chai Wah Wu, Sep 20 2023

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 2, 7, 13, 34, 81, 175, 403, 890, 1977, 4262, 9356, 19963, 42573, 90865, 191206, 401803, 837898, 1744231, 3607504, 7436628, 15254309, 31185686, 63552725, 128963236, 260933000, 526140540, 1057927323, 2120500885, 4239012067, 8449746787, 16799938614

Offset: 0

Gus Wiseman, Dec 16 2019

nonn

Number of sets of disjoint nonempty sets of nonempty multisets of positive integers with total sum n.

			The a(4) = 13 partitions:
  ((4))  ((22))  ((31))      ((211))      ((1111))
                 ((1)(3))    ((1)(21))    ((1)(111))
                 ((1))((3))  ((2)(11))    ((1))((111))
                             ((1))((21))
                             ((2))((11))

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001970, A007713, A050343, A063834, A089259, A261049, A271619, A279375, A294617, A318565, A323787-A323795, A330452-A330459, A330460.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],UnsameQ@@Join@@#&]],{n,0,10}]

\\ here BellP is A000110 as series.
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ Andrew Howroyd, Dec 29 2019

a(n) = Sum_{0 <= k <= n} A330463(n,k) * A000110(k).

Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019

A330463 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 3, 2, 0, 0, 5, 4, 0, 0, 0, 7, 11, 1, 0, 0, 0, 11, 20, 6, 0, 0, 0, 0, 15, 40, 16, 0, 0, 0, 0, 0, 22, 68, 40, 3, 0, 0, 0, 0, 0, 30, 120, 91, 11, 0, 0, 0, 0, 0, 0, 42, 195, 186, 41, 0, 0, 0, 0, 0, 0, 0, 56, 320, 367, 105, 3, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 19 2019

			Triangle begins:
  1
  0  1
  0  2  0
  0  3  2  0
  0  5  4  0  0
  0  7 11  1  0  0
  0 11 20  6  0  0  0
  0 15 40 16  0  0  0  0
  0 22 68 40  3  0  0  0  0
  ...
Row n = 5 counts the following sets of multisets:
  {{5}}          {{1},{4}}        {{1},{2},{1,1}}
  {{1,4}}        {{2},{3}}
  {{2,3}}        {{1},{1,3}}
  {{1,1,3}}      {{1},{2,2}}
  {{1,2,2}}      {{2},{1,2}}
  {{1,1,1,2}}    {{3},{1,1}}
  {{1,1,1,1,1}}  {{1},{1,1,2}}
                 {{1,1},{1,2}}
                 {{2},{1,1,1}}
                 {{1},{1,1,1,1}}
                 {{1,1},{1,1,1}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A261049.
Column k = 1 is A000041.
Multisets of multisets are A061260, with row sums A001970.
Sets of sets are A330462, with row sums A050342.
Multisets of sets are A285229, with row sums A089259.
Sets of disjoint sets are A330460, with row sums A294617.
Cf. A007716, A060016, A063834, A255906, A271619, A279785, A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(
       combinat[numbpart](i), j)*expand(b(n-i*j, i-1)*x^j), j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Dec 30 2019

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,2],And[UnsameQ@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[
     PartitionsP[i], j]*Expand[b[n - i*j, i - 1]*x^j], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000041(j). - Andrew Howroyd, Dec 29 2019

cp's OEIS Frontend

A320328 Number of square multiset partitions of integer partitions of n.

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A358908 Number of finite sequences of distinct integer partitions with total sum n and weakly decreasing lengths.

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Mathematica

PARI

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A296118 Number of ways to choose a factorization of each factor in a strict factorization of n.

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Mathematica

PARI

Formula

A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

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Mathematica

A358906 Number of finite sequences of distinct integer partitions with total sum n.

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Maple

Mathematica

Formula

A387120 Number of ways to choose a different constant integer partition of each prime index of n.

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Mathematica

A320330 Number of T_0 multiset partitions of integer partitions of n.

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Mathematica

A321470 Number of integer partitions of the n-th triangular number 1 + 2 + ... + n that can be obtained by choosing a partition of each integer from 1 to n and combining.

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Mathematica

Python