A001970 -id:A001970 - cp's OEIS frontend

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{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}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]

A316439 Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3

Offset: 1

Gus Wiseman, Jul 03 2018

			The factorizations of 24 are (2*2*2*3), (2*2*6), (2*3*4), (2*12), (3*8), (4*6), (24) so the 24th row is {1, 3, 2, 1}.
Triangle begins:
  {}
  1
  1
  1  1
  1
  1  1
  1
  1  1  1
  1  1
  1  1
  1
  1  2  1
  1
  1  1
  1  1
  1  2  1  1
  1
  1  2  1
  1
  1  2  1
  1  1
  1  1
  1
  1  3  2  1
  1  1
  1  1
  1  1  1
  1  2  1
  1
  1  3  1

Alois P. Heinz, Rows n = 1..20000, flattened

Cf. A001222 (row lengths), A001055 (row sums), A001970, A007716, A045778, A162247, A259936, A281116, A303386.

g:= proc(n, k) option remember; `if`(n>k, 0, x)+
      `if`(isprime(n), 0, expand(x*add(`if`(d>k, 0,
      g(n/d, d)), d=numtheory[divisors](n) minus {1, n})))
    end:
T:= n-> `if`(n=1, [][], (p-> seq(coeff(p, x, i)
        , i=1..degree(p)))(g(n$2))):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 11 2019

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

A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518

Offset: 0

N. J. A. Sloane

An ordinary partition is a row of numbers in nondecreasing order whose sum is n. Here the numbers are in a three-dimensional pile, nondecreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113, ...] and row sums of the convolution triangle A161564. - Gary W. Adamson, Jun 13 2009

			Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(4) = 26 solid partitions, represented as chains of chains of integer partitions:
  ((1))  ((2))       ((3))            ((4))
         ((11))      ((21))           ((22))
         ((1)(1))    ((111))          ((31))
         ((1))((1))  ((2)(1))         ((211))
                     ((11)(1))        ((1111))
                     ((2))((1))       ((2)(2))
                     ((1)(1)(1))      ((3)(1))
                     ((11))((1))      ((21)(1))
                     ((1)(1))((1))    ((11)(11))
                     ((1))((1))((1))  ((111)(1))
                                      ((2))((2))
                                      ((3))((1))
                                      ((2)(1)(1))
                                      ((21))((1))
                                      ((11))((11))
                                      ((11)(1)(1))
                                      ((111))((1))
                                      ((2)(1))((1))
                                      ((1)(1)(1)(1))
                                      ((11)(1))((1))
                                      ((2))((1))((1))
                                      ((1)(1))((1)(1))
                                      ((1)(1)(1))((1))
                                      ((11))((1))((1))
                                      ((1)(1))((1))((1))
                                      ((1))((1))((1))((1))
(End)

P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Suresh Govindarajan, Table of n, a(n) for n = 0..72
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, arXiv:1406.5605 [cond-mat.stat-mech], 2014; J. Stat. Phys. 158 (2015) 950-967.
Suresh Govindarajan, Solid Partitions Project Dec 14, 2010.
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
P. A. MacMahon, Combinatory analysis.
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003; J. Phys. A 36 (2003), no. 24, 6651-6659.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
Eric Weisstein's World of Mathematics, Solid Partition
Wikipedia, Solid partition
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000041, A000219 (2-dim), A000294, A000334 (4-dim), A000390 (5-dim), A002835, A002836, A005980, A037452 (inverse Euler trans.), A080207, A007326, A000416 (6-dim), A000427 (7-dim), A179855 (8-dim).
Cf. A161564. - Gary W. Adamson, Jun 13 2009
Cf. A001970, A002974, A003293, A007713, A114736, A117433, A323657.

planePtns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And@@(GreaterEqual@@@Transpose[PadRight[#]])&],{ptn,IntegerPartitions[n]}];
solidPtns[n_]:=Join@@Table[Select[Tuples[planePtns/@y],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])&],{y,IntegerPartitions[n]}];
Table[Length[solidPtns[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

More terms from the Mustonen and Rajesh article, May 02 2003
a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010
a(63)-a(68) found by Suresh Govindarajan and students, Jun 01 2011
a(69)-a(72) found by Suresh Govindarajan and Srivatsan Balakrishnan, Jan 03 2013

A050336 Number of ways of factoring n with one level of parentheses.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 14, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 27, 3, 3, 3, 31, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 57, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 58, 3, 12, 1, 9, 3, 12, 1, 83, 1, 3, 9, 9, 3, 12, 1, 57, 14, 3, 1, 41, 3, 3, 3, 23, 1, 41, 3, 9

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2*2) = (3*2)*(2) = (3)*(2*2) = (3)*(2)*(2).

R. J. Mathar, Table of n, a(n) for n = 1..2303

Cf. A001055, A050337, A050338, A050339, A050340, A050341.

a(p^k)=A001970. a(A002110)=A000258.

Dirichlet g.f.: Product_{n>=2}(1/(1-1/n^s)^A001055(n)).

a(n) = A050337(A101296(n)). - R. J. Mathar, May 26 2017

A072706 Number of unimodal partitions/compositions of n into distinct terms.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131

Offset: 0

Henry Bottomley, Jul 04 2002

nonn

Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019

			a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
From _Joerg Arndt_, Mar 25 2014: (Start)
The a(10) = 33 such compositions of 10 are:
01:  [ 1 2 3 4 ]
02:  [ 1 2 4 3 ]
03:  [ 1 2 7 ]
04:  [ 1 3 4 2 ]
05:  [ 1 3 6 ]
06:  [ 1 4 3 2 ]
07:  [ 1 4 5 ]
08:  [ 1 5 4 ]
09:  [ 1 6 3 ]
10:  [ 1 7 2 ]
11:  [ 1 9 ]
12:  [ 2 3 4 1 ]
13:  [ 2 3 5 ]
14:  [ 2 4 3 1 ]
15:  [ 2 5 3 ]
16:  [ 2 7 1 ]
17:  [ 2 8 ]
18:  [ 3 4 2 1 ]
19:  [ 3 5 2 ]
20:  [ 3 6 1 ]
21:  [ 3 7 ]
22:  [ 4 3 2 1 ]
23:  [ 4 5 1 ]
24:  [ 4 6 ]
25:  [ 5 3 2 ]
26:  [ 5 4 1 ]
27:  [ 6 3 1 ]
28:  [ 6 4 ]
29:  [ 7 2 1 ]
30:  [ 7 3 ]
31:  [ 8 2 ]
32:  [ 9 1 ]
33:  [ 10 ]
(End)

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

The non-strict version is A001523.

Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 2014

b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
Table[If[n==0,1,Sum[2^(Length[ptn]-1),{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,15}] (* Gus Wiseman, Dec 31 2019 *)

N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014

a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).
G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003
G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019

A319066 Number of partitions of integer partitions of n where all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 59, 82, 128, 176, 273, 371, 553, 768, 1119, 1544, 2235, 3084, 4410, 6111, 8649, 11982, 16901, 23383, 32780, 45396, 63365, 87622, 121946, 168407, 233605, 322269, 445723, 613922, 847131, 1164819, 1603431, 2201370, 3023660, 4144124, 5680816

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{1,1},{1,1}}      {{1,1,1,2}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

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

Cf. A001970, A047968, A261049, A279787, A305551, A306017, A319056.

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],SameQ@@Length/@#&]],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k*y + O(x*x^n))); concat([1], sum(k=1, n, EulerT(Vec(polcoef(p, k, y), -n))))} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (2),(1)    (1,1,2)
                  (1,1),(1)  (1,2,1)
                             (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1,1,1),(1)
(End)

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.
The non-strict version is A075900.
Starting with a reversed partition gives A323583.
Starting with a partition gives A336134.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings with distinct sums are A336127.
Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

G.f.: Product_{k>=1} (1 + A011782(k)*x^k).

a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019

A319721 Number of non-isomorphic antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 4, 8, 24, 50, 148, 349, 1014, 2717, 8114

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001055, A001970, A007716, A096827, A253249, A285572, A285573, A293993, A318099, A319616-A319646, A319719.

A358914 Number of twice-partitions of n into distinct strict partitions.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 13, 20, 32, 51, 83, 130, 206, 320, 496, 759, 1171, 1786, 2714, 4104, 6193, 9286, 13920, 20737, 30865, 45721, 67632, 99683, 146604, 214865, 314782, 459136, 668867, 972425, 1410458, 2040894, 2950839, 4253713, 6123836, 8801349, 12627079

Offset: 0

Gus Wiseman, Dec 11 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) = 13 twice-partitions:
  ((1))  ((2))  ((3))     ((4))      ((5))      ((6))
                ((21))    ((31))     ((32))     ((42))
                ((2)(1))  ((3)(1))   ((41))     ((51))
                          ((21)(1))  ((3)(2))   ((321))
                                     ((4)(1))   ((4)(2))
                                     ((21)(2))  ((5)(1))
                                     ((31)(1))  ((21)(3))
                                                ((31)(2))
                                                ((3)(21))
                                                ((32)(1))
                                                ((41)(1))
                                                ((3)(2)(1))
                                                ((21)(2)(1))

Andrew Howroyd, Table of n, a(n) for n = 0..100

The unordered version is A050342, non-strict A261049.
This is the distinct case of A270995.
The case of strictly decreasing sums is A279785.
The case of constant sums is A279791.
For distinct instead of weakly decreasing sums we have A336343.
This is the twice-partition case of A358913.
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, A000219, A075900, A271619, A296122, A304969, A321449, A336342, A358901, A358906, A358907.

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

seq(n,k)={my(u=Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))-1)); Vec(prod(k=1, n, my(c=u[k]); sum(j=0, min(c,n\k), x^(j*k)*c!/(c-j)!,  O(x*x^n))))} \\ Andrew Howroyd, Dec 31 2022

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

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 12, 12, 17, 20, 22, 28, 35, 39, 48, 55, 65, 79, 90, 105, 121, 143, 166, 190, 219, 254, 290, 332, 382, 436, 493, 567, 637, 729, 824, 931, 1052, 1186, 1334, 1504, 1691, 1894, 2123, 2380, 2664, 2968, 3319, 3704, 4119, 4586, 5110

Offset: 0

Gus Wiseman, May 26 2018

nonn

			The a(9) = 7 integer partitions are (9), (72), (54), (522), (333), (3222), (111111111).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..360 (terms 0..300 from Alois P. Heinz)

Cf. A000837, A001055, A001970, A007359, A007716, A051424, A078374, A100953, A285572, A285573, A302696, A303362, A305079, A305149, A305150.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,20}]

More terms from Alois P. Heinz, May 26 2018

cp's OEIS Frontend

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

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A316439 Irregular triangle where T(n,k) is the number of factorizations of n into k factors > 1, with k ranging from 1 to Omega(n).

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A000293 a(n) = number of solid (i.e., three-dimensional) partitions of n.

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A050336 Number of ways of factoring n with one level of parentheses.

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A072706 Number of unimodal partitions/compositions of n into distinct terms.

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A319066 Number of partitions of integer partitions of n where all parts have the same length.

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A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

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A319721 Number of non-isomorphic antichains of multisets of weight n.