A213427 -id:A213427 - cp's OEIS frontend

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

Offset: 0

Gus Wiseman, Mar 20 2025

			The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
The a(1) = 1 through a(12) = 13 partitions:
  1  2  3  4    5    6     7    8      9      A      B      C
           211  221  411   322  332    441    433    443    552
                311  2211  331  422    522    442    533    633
                           511  611    711    622    551    822
                                3311   42111  811    722    A11
                                32111         3322   911    4422
                                              4411   42221  5511
                                              32221  53111  33321
                                              43111  62111  52221
                                              52111         54111
                                                            63111
                                                            72111
                                                            3222111

Normal multiset partitions of this type are counted by A116539, see A381718.
These partitions are ranked by A293511.
MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
For distinct block-sums instead of blocks we have A382460, ranked by A381870.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

Original entry on oeis.org

1, 1, 2, 7, 39, 236, 1836, 16123, 162008, 1802945, 22012335, 291290460, 4144907830, 62986968311, 1016584428612, 17344929138791, 311618472138440, 5875109147135658, 115894178676866576, 2385755803919949337, 51133201045333895149, 1138659323863266945177, 26296042933904490636133

Offset: 0

Gus Wiseman, Dec 22 2019

nonn

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

			Non-isomorphic representatives of the a(4) = 39 trees, with singleton leaves (x) replaced by just x:
  (1111)      (1112)      (1122)      (1123)      (1234)
  (1(111))    (1(112))    (1(122))    (1(123))    (1(234))
  (11(11))    (11(12))    (11(22))    (11(23))    (12(34))
  ((11)(11))  (12(11))    (12(12))    (12(13))    ((12)(34))
  (1(1(11)))  (2(111))    ((11)(22))  (2(113))    (1(2(34)))
              ((11)(12))  (1(1(22)))  (23(11))
              (1(1(12)))  ((12)(12))  ((11)(23))
              (1(2(11)))  (1(2(12)))  (1(1(23)))
              (2(1(11)))              ((12)(13))
                                      (1(2(13)))
                                      (2(1(13)))
                                      (2(3(11)))

The case with all atoms equal or all atoms different is A000669.
Not requiring singleton-reduction gives A330465.
Labeled versions are A316651 (normal orderless) and A330471 (strongly normal).
The case where the leaves are sets is A330626.
Row sums of A339645.
Cf. A000311, A005121, A005804, A141268, A213427, A292504, A292505, A318812, A318848, A318849, A330467, A330469, A330474, A330624.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n] = polcoef( sEulerT(x*Ser(v[1..n])), n )); x*Ser(v)}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(7) and beyond from Andrew Howroyd, Dec 11 2020

A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 12, 138, 2652, 78106, 3256404, 182463296, 13219108288, 1202200963522, 134070195402644, 17989233145940910, 2858771262108762492, 530972857546678902490, 113965195745030648131036, 27991663753030583516229824, 7800669355870672032684666900, 2448021231611414334414904013956

Offset: 0

Gus Wiseman, Dec 27 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			The a(0) = 1 through a(3) = 12 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,2}}
                  {{2},{1,3}}
                  {{3},{1,2}}

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

Row sums of A330776.
The unlabeled version is A330474.
The strongly normal case is A330475.
The tree version is A330654.
The maximum-depth case is A330676.
The case where the atoms are all different is A005121.
The case where the atoms are all equal is A318813.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000311, A000669, A001678, A213427, A318812, A330675, A330679.

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[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2019

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2019

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 04 2020

This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.

			Triangle begins:
   1:          16: 0 1 3 2    31: 1            46: 0 1
   2: 1        17: 1          32: 0 1 5 8 4    47: 1
   3: 1        18: 0 1 2      33: 0 1          48: 0 1 10 23 15
   4: 0 1      19: 1          34: 0 1          49: 0 1
   5: 1        20: 0 1 2      35: 0 1          50: 0 1 2
   6: 0 1      21: 0 1        36: 0 1 7 7      51: 0 1
   7: 1        22: 0 1        37: 1            52: 0 1 2
   8: 0 1 1    23: 1          38: 0 1          53: 1
   9: 0 1      24: 0 1 5 5    39: 0 1          54: 0 1 5 5
  10: 0 1      25: 0 1        40: 0 1 5 5      55: 0 1
  11: 1        26: 0 1        41: 1            56: 0 1 5 5
  12: 0 1 2    27: 0 1 1      42: 0 1 3        57: 0 1
  13: 1        28: 0 1 2      43: 1            58: 0 1
  14: 0 1      29: 1          44: 0 1 2        59: 1
  15: 0 1      30: 0 1 3      45: 0 1 2        60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
  ()  (6*8)      (2*3*8)->(6*8)       (2*2*2*6)->(2*4*6)->(6*8)
      (2*24)     (2*4*6)->(6*8)       (2*2*3*4)->(2*3*8)->(6*8)
      (3*16)     (2*3*8)->(2*24)      (2*2*3*4)->(2*4*6)->(6*8)
      (4*12)     (2*3*8)->(3*16)      (2*2*2*6)->(2*4*6)->(2*24)
      (2*3*8)    (2*4*6)->(2*24)      (2*2*2*6)->(2*4*6)->(4*12)
      (2*4*6)    (2*4*6)->(4*12)      (2*2*3*4)->(2*3*8)->(2*24)
      (3*4*4)    (3*4*4)->(3*16)      (2*2*3*4)->(2*3*8)->(3*16)
      (2*2*12)   (3*4*4)->(4*12)      (2*2*3*4)->(2*4*6)->(2*24)
      (2*2*2*6)  (2*2*12)->(2*24)     (2*2*3*4)->(2*4*6)->(4*12)
      (2*2*3*4)  (2*2*12)->(4*12)     (2*2*3*4)->(3*4*4)->(3*16)
                 (2*2*2*6)->(6*8)     (2*2*3*4)->(3*4*4)->(4*12)
                 (2*2*3*4)->(6*8)     (2*2*2*6)->(2*2*12)->(2*24)
                 (2*2*2*6)->(2*24)    (2*2*2*6)->(2*2*12)->(4*12)
                 (2*2*2*6)->(4*12)    (2*2*3*4)->(2*2*12)->(2*24)
                 (2*2*3*4)->(2*24)    (2*2*3*4)->(2*2*12)->(4*12)
                 (2*2*3*4)->(3*16)
                 (2*2*3*4)->(4*12)
                 (2*2*2*6)->(2*4*6)
                 (2*2*3*4)->(2*3*8)
                 (2*2*3*4)->(2*4*6)
                 (2*2*3*4)->(3*4*4)
                 (2*2*2*6)->(2*2*12)
                 (2*2*3*4)->(2*2*12)

Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upfacs[y],y],{y,facs[n]}],{n},First[facs[n]]],Length[#]==k-1&]],{n,100},{k,PrimeOmega[n]}]

T(2^n,k) = A330785(n,k).

T(n,1) + T(n,2) = 1.

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 16, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 16, 1, 3, 1, 2, 1, 3, 1, 27, 1, 1, 2, 2, 1, 3, 1, 16, 2, 1, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 27 2019

nonn

First differs from A317145 at a(32) = 5, A317145(32) = 4.
A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
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.
Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

			The a(n) multisystems for n = 2, 6, 12, 24, 48:
  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}
              {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}
                           {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}
                           {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}
                           {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}
                                                {{{{1}}},{{{1}},{{2},{1,1}}}}
                                                {{{{1}}},{{{1,2}},{{1},{1}}}}
                                                {{{{1},{1}}},{{{2}},{{1,1}}}}
                                                {{{{1},{1,2}}},{{{1}},{{1}}}}
                                                {{{{1,1}}},{{{2}},{{1},{1}}}}
                                                {{{{1}}},{{{2}},{{1},{1,1}}}}
                                                {{{{1},{2}}},{{{1}},{{1,1}}}}
                                                {{{{1,2}}},{{{1}},{{1},{1}}}}
                                                {{{{2}}},{{{1}},{{1},{1,1}}}}
                                                {{{{2}}},{{{1,1}},{{1},{1}}}}
                                                {{{{2},{1,1}}},{{{1}},{{1}}}}

The last nonzero term in row n of A330667 is a(n).
The chain version is A317145.
The non-maximal version is A318812.
Unlabeled versions are A330664 and A330663.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

a(2^n) = A000111(n - 1).

a(product of n distinct primes) = A006472(n).

A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 4, 15, 11, 20, 21, 90, 51, 80, 32, 468, 166, 2910, 124, 521, 277, 20644, 266, 621, 1761, 1866, 841, 165874, 1374, 1484344, 436, 3797, 12741, 5383, 3108, 14653890, 103783, 31323, 2294, 158136988, 12419, 1852077284, 6382, 20786, 939131, 23394406084

Offset: 1

Gus Wiseman, Sep 04 2018

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(12) = 21 multisystems on {1,1,2,3} (commas elided):
  {1123}  {{1}{123}}  {{1}{1}{23}}  {{{1}}{{1}{23}}}
          {{2}{113}}  {{1}{2}{13}}  {{{23}}{{1}{1}}}
          {{3}{112}}  {{1}{3}{12}}  {{{1}}{{2}{13}}}
          {{11}{23}}  {{2}{3}{11}}  {{{2}}{{1}{13}}}
          {{12}{13}}                {{{13}}{{1}{2}}}
                                    {{{1}}{{3}{12}}}
                                    {{{3}}{{1}{12}}}
                                    {{{12}}{{1}{3}}}
                                    {{{2}}{{3}{11}}}
                                    {{{3}}{{2}{11}}}
                                    {{{11}}{{2}{3}}}

Row sums of A330727.
Cf. A001055, A002846, A005121, A181821, A213427, A281118, A281119, A305936, A318762, A318812, A318813.
Cf. A318847, A318848, A318849.

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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
tmsp[m_]:=Prepend[Join@@Table[tmsp[c],{c,Select[mps[m],1

a(n) = A318812(A181821(n)).
a(prime(n)) = A318813(n).
a(2^n) = A005121(n).

Terminology corrected by Gus Wiseman, Jan 04 2020

More terms from Jinyuan Wang, Jun 26 2020

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

A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

Original entry on oeis.org

2, 6, 8, 30, 36, 40, 48, 64, 210, 252, 270, 280, 300, 324, 336, 360, 400, 432, 448, 480, 576, 640, 768, 1024, 2310, 2772, 2940, 2970, 3080, 3150, 3300, 3528, 3564, 3696, 3780, 3920, 3960, 4050, 4200, 4400, 4500, 4536, 4704, 4752, 4860, 4928, 5040, 5280, 5400

Offset: 1

Gus Wiseman, Nov 13 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

These partitions are those that are finer than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

			The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (21), (111), (321), (2211), (3111), (21111), (111111), (4321), (42211), (32221), (43111), (33211), (222211), (421111), (322111), (331111), (2221111), (4111111), (3211111), (22111111), (31111111), (211111111), (1111111111).
The partition (322111) has Heinz number 360 and can be partitioned as ((1)(2)(3)(112)), ((1)(2)(12)(13)), or ((1)(11)(3)(22)), so 360 belongs to the sequence.

Subsequence of A242422.
Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
Cf. A321467, A321468, A321470, A321472, A321514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[2,1000],Select[Map[Total[primeMS[#]]&,facs[#],{2}],Sort[#]==Range[Max@@#]&]!={}&]

A321472 Heinz numbers of integer partitions whose parts can be further partitioned and flattened to obtain the partition (k, ..., 3, 2, 1) for some k.

Original entry on oeis.org

2, 5, 6, 13, 21, 22, 25, 29, 30, 46, 47, 57, 73, 85, 86, 91, 102, 107, 121, 123, 130, 142, 147, 151, 154, 165, 175, 185, 197, 201, 206, 210, 217, 222, 257, 298, 299

Offset: 1

Gus Wiseman, Nov 13 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

These partitions are those that are coarser than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

			The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (3), (2,1), (6), (4,2), (5,1), (3,3), (10), (3,2,1), (9,1), (15), (8,2), (21), (7,3), (14,1), (6,4), (7,2,1), (28), (5,5), (13,2), (6,3,1), (20,1), (4,4,2), (36), (5,4,1), (5,3,2), (4,3,3), (12,3), (45), (19,2), (27,1), (4,3,2,1).

Subsequence of A242422.
Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
Cf. A321467, A321468, A321470, A321471, A321514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,200],Select[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[#]],Sort[#]==Range[Max@@#]&]!={}&]

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

cp's OEIS Frontend

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

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A330470 Number of non-isomorphic series/singleton-reduced rooted trees on a multiset of size n.

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A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

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A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

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A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

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A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

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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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A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

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