A296122 -id:A296122 - cp's OEIS frontend

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

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}]

A358907 Number of finite sequences of distinct integer compositions with total sum n.

Original entry on oeis.org

1, 1, 2, 8, 18, 54, 156, 412, 1168, 3200, 8848, 24192, 66632, 181912, 495536, 1354880, 3680352, 9997056, 27093216, 73376512, 198355840, 535319168, 1443042688, 3884515008, 10445579840, 28046885824, 75225974912, 201536064896, 539339293824, 1441781213952

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

			The a(1) = 1 through a(4) = 18 sequences:
  ((1))  ((2))   ((3))      ((4))
         ((11))  ((12))     ((13))
                 ((21))     ((22))
                 ((111))    ((31))
                 ((1)(2))   ((112))
                 ((2)(1))   ((121))
                 ((1)(11))  ((211))
                 ((11)(1))  ((1111))
                            ((1)(3))
                            ((3)(1))
                            ((1)(12))
                            ((11)(2))
                            ((1)(21))
                            ((12)(1))
                            ((2)(11))
                            ((21)(1))
                            ((1)(111))
                            ((111)(1))

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

For sets instead of sequences we have A098407, partitions A261049.
This is the strict case of A133494.
The case of distinct sums is A336127, constant sums A074854.
The version for sequences of partitions is A358906.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A218482 counts sequences of compositions with weakly decreasing lengths.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all different Omegas.
A358914 counts twice-partitions into distinct strict partitions.
Cf. A000009, A000041, A000219, A055887, A075900, A296122, A304961, A307068, A336342, A358836, A358912.

g:= proc(n) option remember; ceil(2^(n-1)) end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, (t->
      add(binomial(t, j)*b(n-i*j, i-1, p+j), j=0..min(t, n/i)))(g(i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Dec 15 2022

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

a(16)-a(29) from Alois P. Heinz, Dec 15 2022

A296121 Number of twice-factorizations of n with no repeated factorizations.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 10, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 20, 3, 3, 3, 25, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 47, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 40, 3, 12, 1, 8, 3, 12, 1, 68, 1, 3, 8, 8, 3, 12, 1, 47, 10

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

From Robert G. Wilson v, Dec 05 2017: (Start)
a(n) = 1 iff n equals 1 or is a prime;
a(n) = 2 iff n is a prime squared;
a(n) = 3 iff n is a squarefree semiprime;
a(n) = 5 iff n is a prime cube;
a(n) = 8 iff n is of the form p^2*q, etc.
(End)

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

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A001055, A045778, A050345, A063834, A089723, A281113, A296118, A296119, A296120, A296122.

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

A358833 Number of rectangular twice-partitions of n of type (P,R,P).

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n, so these are twice-partitions of n into partitions with constant lengths and constant sums.

			The a(1) = 1 through a(5) = 8 twice-partitions:
  (1)  (2)     (3)        (4)           (5)
       (11)    (21)       (22)          (32)
       (1)(1)  (111)      (31)          (41)
               (1)(1)(1)  (211)         (221)
                          (1111)        (311)
                          (2)(2)        (2111)
                          (11)(11)      (11111)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the rectangular case of A279787.
This is the case of A306319 with constant sums.
For distinct instead of constant lengths and sums we have A358832.
The version for multiset partitions of integer partitions is A358835.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A281145 counts same-trees.
Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 10, 20, 29, 58, 83, 150, 230, 399, 605, 1037, 1545, 2547, 3879, 6241, 9437, 15085, 22622, 35493, 53438, 82943, 124157, 191267, 284997, 434634, 647437, 979293, 1452182, 2185599, 3228435, 4826596, 7112683, 10575699, 15530404, 22990800, 33651222

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Also the number of odd-length twice-partitions of n into strict partitions.

			The a(1) = 1 through a(6) = 10 twice-partitions with all odd parts:
  (1)  (11)  (3)        (31)        (5)              (33)
             (111)      (1111)      (311)            (51)
             (1)(1)(1)  (11)(1)(1)  (11111)          (3111)
                                    (3)(1)(1)        (111111)
                                    (11)(11)(1)      (3)(11)(1)
                                    (111)(1)(1)      (31)(1)(1)
                                    (1)(1)(1)(1)(1)  (11)(11)(11)
                                                     (111)(11)(1)
                                                     (1111)(1)(1)
                                                     (11)(1)(1)(1)(1)
The a(1) = 1 through a(6) = 10 twice-partitions into strict partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (21)       (31)       (32)             (42)
            (1)(1)(1)  (2)(1)(1)  (41)             (51)
                                  (2)(2)(1)        (321)
                                  (3)(1)(1)        (2)(2)(2)
                                  (21)(1)(1)       (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (21)(2)(1)
                                                   (31)(1)(1)
                                                   (2)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the odd-length case of A270995.
Requiring odd sums also gives A279374 aerated.
This is the case of A358824 with all odd parts.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
A358334 counts twice-partitions into odd-length partitions.
Cf. A000041, A001970, A072233, A271619, A279785, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Flatten[#]]&]],{n,0,10}]

R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)) - 1)); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 31 2022

G.f.: ((1/Product_{k>=1} (1-A000009(k)*x^k)) - (1/Product_{k>=1} (1+A000009(k)*x^k)))/2. - Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130

Offset: 1

Gus Wiseman, Aug 26 2025

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
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. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A358832 Number of twice-partitions of n into partitions of distinct lengths and distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 7, 15, 25, 49, 79, 154, 248, 453, 748, 1305, 2125, 3702, 5931, 9990, 16415, 26844, 43246, 70947, 113653, 182314, 292897, 464614, 739640, 1169981, 1844511, 2888427, 4562850, 7079798, 11064182, 17158151, 26676385, 41075556, 63598025, 97420873, 150043132

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)      (4)       (5)
       (11)  (21)     (22)      (32)
             (111)    (31)      (41)
             (11)(1)  (211)     (221)
                      (1111)    (311)
                      (21)(1)   (2111)
                      (111)(1)  (11111)
                                (21)(2)
                                (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)

This is the case of A271619 with distinct lengths.
These multiset partitions are ranked by A326535 /\ A326533.
This is the case of A358830 with distinct sums.
For constant instead of distinct lengths and sums we have A358833.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A306319, A358334, A358836.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Total/@#&&UnsameQ@@Length/@#&]],{n,0,10}]

seq(n)={ local(Cache=Map());
  my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
  my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
  z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m-1,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

A358834 Number of odd-length twice-partitions of n into odd-length partitions.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 11, 24, 35, 74, 109, 213, 336, 624, 986, 1812, 2832, 5002, 7996, 13783, 21936, 37528, 59313, 99598, 158356, 262547, 415590, 684372, 1079576, 1759984, 2779452, 4491596, 7069572, 11370357, 17841534, 28509802, 44668402, 70975399, 110907748

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (111)      (211)      (221)            (222)
            (1)(1)(1)  (2)(1)(1)  (311)            (321)
                                  (11111)          (411)
                                  (2)(2)(1)        (21111)
                                  (3)(1)(1)        (2)(2)(2)
                                  (111)(1)(1)      (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (111)(2)(1)
                                                   (211)(1)(1)
                                                   (2)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for set partitions is A003712.
If the parts are also odd we get A279374.
The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
This is the odd-length case of A358334.
This is the odd-lengths case of A358824.
For odd sums instead of lengths we have A358826.
The case of odd sums also is the bisection of A358827.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A270995, A271619, A279785, A358823, A358837.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {2}     {3}        {4}           {5}              {6}
       {11}    {12}       {13}          {14}             {15}
       {1}{1}  {111}      {22}          {23}             {24}
               {1}{1}{1}  {112}         {113}            {33}
                          {1111}        {122}            {114}
                          {2}{2}        {1112}           {123}
                          {11}{11}      {11111}          {222}
                          {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
                                                         {1122}
                                                         {3}{3}
                                                         {11112}
                                                         {111111}
                                                         {12}{12}
                                                         {2}{2}{2}
                                                         {111}{111}
                                                         {11}{11}{11}
                                                         {1}{1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For just constant sums we have A305551, ranked by A326534.
For just constant lengths we have A319066, ranked by A320324.
The version for set partitions is A327899.
For distinct instead of constant lengths and sums we have A358832.
The version for twice-partitions is A358833.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A007425, A141199, A327908, A356932, A358831.

Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]

P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022

cp's OEIS Frontend

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

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

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A358907 Number of finite sequences of distinct integer compositions with total sum n.

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A296121 Number of twice-factorizations of n with no repeated factorizations.

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A358833 Number of rectangular twice-partitions of n of type (P,R,P).

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A358823 Number of odd-length twice-partitions of n into partitions with all odd parts.

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A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

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A358832 Number of twice-partitions of n into partitions of distinct lengths and distinct sums.

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