A055887 -id:A055887 - cp's OEIS frontend

A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 36, 39, 40, 41, 44, 45, 47, 48, 50, 51, 52, 54, 55, 59, 60, 62, 64, 65, 66, 67, 68, 72, 75, 78, 80, 81, 82, 83, 85, 88, 90, 93, 94, 96, 99, 100, 102, 104, 108

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

The initial version is A356940.
Intervals are counted by A000012, A001227, ranked by A073485.
Other types: A107742, A356936, A356937, A356938.
Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers.
A001055 counts factorizations.
A001221 counts prime divisors, sum A001414.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Select[Range[100],And@@chQ/@primeMS/@primeMS[#]&]

A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 37, 76, 159, 326, 671, 1376, 2815, 5759, 11774, 24083, 49249, 100632, 205490, 419420, 855799, 1745889, 3561867, 7268240, 14836127, 30295633, 61888616

Offset: 0

Gus Wiseman, Mar 26 2025

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(4) = 9 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
                    {{1},{2,2}}    {{1,1},{2,2}}
                    {{1},{2},{3}}  {{1},{2,2,2}}
                                   {{2},{1,1,1}}
                                   {{1},{2},{2,2}}
                                   {{1},{2},{3,3}}
                                   {{1},{3},{2,2}}
                                   {{1},{2},{3},{4}}
The a(5) = 19 factorizations:
  32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
      4*8   2*4*9    2*3*5*9
      2*81  2*3*8    2*3*5*49
      4*27  2*3*125  2*3*7*25
      9*8   2*9*25
      3*16  2*5*27
            5*4*9

Without distinct sums we have A055887.
Twice-partitions of this type are counted by A279786.
For distinct blocks instead of sums we have A304969.
Without constant blocks we have A326519.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A381718.
For equal instead of distinct block-sums we have A382204.
For equal block-sums and strict blocks we have A382429.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count multiset partitions of prime indices, strict A045778.
A089259 counts set multipartitions of integer partitions.
A321469 counts multiset partitions with distinct block-sums, ranks A326535.
Normal multiset partitions: A035310, A116540, A255906, A317532.
Set multipartitions with distinct sums: A279785, A381806, A381870.
Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(14)-a(26) from Christian Sievers, Apr 04 2025

A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

Original entry on oeis.org

1, -1, 0, 1, -1, 1, -1, 1, 0, -2, 4, -4, 1, 3, -5, 4, -3, 3, -1, -6, 13, -12, 2, 9, -13, 10, -6, 6, -4, -9, 28, -30, 5, 25, -28, 5, 9, 7, -27, 11, 32, -47, 2, 51, -27, -74, 128, -34, -131, 183, -78, -15, -37, 97, 89, -480, 649, -242, -498, 904, -663, 223, -140, 169, 488, -1818

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

sign

Invert transform of A010815.

Alternating row sums of Riordan triangle (1, 1 - Product_{j>=1} (1-x^j) ), See A341418(n, m) without column {1, repeat(0)} for m = 0 and n >= 0. - Wolfdieter Lang, Feb 17 2021

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A010815, A055887, A299105, A341418.

Cf. A307057, A307058, A307060, A307062, A307063.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 - x^j: j in [1..m+2]])) )); // G. C. Greubel, Sep 08 2023

nmax=65; CoefficientList[Series[1/(2 - Product[(1 - x^k), {k, nmax}]), {x, 0, nmax}], x]

from sage.modular.etaproducts import qexp_eta
m=80;
def f(x): return 1/(2 - qexp_eta(QQ[['q']], m+2).subs(q=x) )
def A307059_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307059_list(m) # G. C. Greubel, Sep 08 2023

a(0) = 1; a(n) = Sum_{k=1..n} A010815(k)*a(n-k).

G.f.: 1/(2 - QPochhammer(x)). - G. C. Greubel, Sep 08 2023

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

A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 22, 23, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 86, 87, 92, 93, 94, 95, 112, 113, 114, 115, 116, 117, 118, 119

Offset: 0

Gus Wiseman, Mar 27 2024

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 binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The terms together with their prime indices of binary indices begin:
   0: {}
   1: {{}}
   2: {{1}}
   3: {{},{1}}
   6: {{1},{2}}
   7: {{},{1},{2}}
   8: {{1,1}}
   9: {{},{1,1}}
  10: {{1},{1,1}}
  11: {{},{1},{1,1}}
  12: {{2},{1,1}}
  13: {{},{2},{1,1}}
  14: {{1},{2},{1,1}}
  15: {{},{1},{2},{1,1}}
  22: {{1},{2},{3}}
  23: {{},{1},{2},{3}}
  28: {{2},{1,1},{3}}
  29: {{},{2},{1,1},{3}}
  30: {{1},{2},{1,1},{3}}
  31: {{},{1},{2},{1,1},{3}}
  32: {{1,2}}

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

The case with squarefree product of prime indices is A371293.
For binary indices of each prime index we have A371447, A371448.
The connected components of this multiset system are counted by A371452.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A001222, A055887, A255906, A326782, A368109, A371291, A371294.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],normQ[Join@@prix/@bpe[#]]&]

from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    for n in count(0):
        s = set()
        b = [(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1']
        for i in b:
            p = factorint(i)
            for j in p:
                s.add(sieve.search(j)[0])
        x = sorted(s)
        y = len(x)
        if sum(x) == (y*(y+1))//2:
            yield n
A371292_list = list(islice(a_gen(), 65)) # John Tyler Rascoe, May 21 2024

A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 12, 21, 38, 68, 120, 212, 377, 670, 1188, 2107, 3740, 6638, 11778, 20898, 37084, 65808, 116775, 207212, 367696, 652478, 1157815, 2054524, 3645730, 6469316, 11479734, 20370656, 36147506, 64143372, 113821732, 201975429, 358403220, 635982680, 1128544452, 2002589998

Offset: 0

Ilya Gutkovskiy, Mar 21 2019

nonn

Invert transform of A000700.

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A000700, A055887, A302017, A304969, A307059.

Row sums of A341279.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 + x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 09 2021

nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

m=80;
def f(x): return 1/(2 - product(1+x^(2*j-1) for j in range(1,m+3)))
def A307058_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307058_list(m) # G. C. Greubel, Jan 24 2024

a(0) = 1; a(n) = Sum_{k=1..n} A000700(k)*a(n-k).
From G. C. Greubel, Jan 24 2024: (Start)
G.f.: (1+x)/(2*(1+x) - x*QPochhammer(-1/x; x^2)).
G.f.: 1/( 2 - x^(1/24)*etx(x^2)^2/(eta(x^4)*eta(x)) ), where eta(x) is the Dedekind eta function. (End)

A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 26, 39, 78, 142, 320, 488, 913, 1558, 2798, 5865, 9482, 16742, 28474, 50814, 82800, 172540, 266093, 472432, 790824, 1361460, 2251665, 3844412, 7205416, 11370048, 19483502, 32416924, 54367066, 88708832, 149179800, 239738369, 445689392

Offset: 0

Gus Wiseman, Jul 19 2020

nonn

A strict composition of n (A032020) is a finite sequence of distinct positive integers summing to n.

Is there a simple generating function?

			The a(1) = 1 through a(5) = 11 ways:
  (1)  (2)  (3)      (4)        (5)
            (2,1)    (3,1)      (3,2)
            (1),(2)  (1),(3)    (4,1)
            (2),(1)  (3),(1)    (1),(4)
                     (1),(2,1)  (2),(3)
                     (2,1),(1)  (3),(2)
                                (4),(1)
                                (1),(3,1)
                                (2,1),(2)
                                (2),(2,1)
                                (3,1),(1)

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

Multiset partitions of partitions are A001970.
Strict compositions are counted by A032020, A072574, and A072575.
Splittings of strict partitions are A072706.
Set partitions of strict partitions are A294617.
Splittings of partitions with distinct sums are A336131.
Cf. A008289, A011782, A304786, A318683, A318684, A319794, A323583, A336128, A336130, A336132, A336133.
Partitions:
- Partitions of each part of a partition are A063834.
- Compositions of each part of a partition are A075900.
- Strict partitions of each part of a partition are A270995.
- Strict compositions of each part of a partition are A336141.
Strict partitions:
- Partitions of each part of a strict partition are A271619.
- Compositions of each part of a strict partition are A304961.
- Strict partitions of each part of a strict partition are A279785.
- Strict compositions of each part of a strict partition are A336142.
Compositions:
- Partitions of each part of a composition are A055887.
- Compositions of each part of a composition are A133494.
- Strict partitions of each part of a composition are A304969.
- Strict compositions of each part of a composition are A307068.
Strict compositions:
- Partitions of each part of a strict composition are A336342.
- Compositions of each part of a strict composition are A336127.
- Strict partitions of each part of a strict composition are A336343.
- Strict compositions of each part of a strict composition are A336139.

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

\\ here Q(N) gives A000009 as a vector.
Q(n) = {Vec(eta(x^2 + O(x*x^n))/eta(x + O(x*x^n)))}
seq(n)={my(b=Q(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, 1 + y*x^k*b[1+k] + O(x*x^n)))]} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} k! * [y^k](Product_{j>=1} 1 + y*x^j*A000009(j)). - Andrew Howroyd, Apr 16 2021

A356934 Number of multisets of odd-size multisets whose multiset union is a size-n multiset covering an initial interval with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 2, 6, 17, 46, 166, 553, 2093

Offset: 0

Gus Wiseman, Sep 09 2022

			The a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,1,2}}      {{1},{1,1,2}}
                    {{1,2,3}}      {{1},{1,2,2}}
                    {{1},{1},{1}}  {{1},{1,2,3}}
                    {{1},{1},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,1,1}}
                                   {{2},{1,1,2}}
                                   {{2},{1,1,3}}
                                   {{2},{1,3,4}}
                                   {{3},{1,1,2}}
                                   {{3},{1,2,4}}
                                   {{4},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A035310, A063834, A330783, A356938, A356943, A356954.
Other types: A050330, A356932, A356933, A356935.
Cf. A055887, A072233, A270995, A304969, A349050, A349055.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],OddQ[Times@@Length/@#]&]],{n,0,5}]

A356937 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 3, 9, 29, 94, 310, 1026, 3411, 11360, 37886, 126442, 422203, 1410189, 4711039, 15740098, 52593430, 175742438, 587266782, 1962469721, 6558071499, 21915580437, 73237274083, 244744474601, 817889464220, 2733235019732, 9133973730633, 30524096110942, 102006076541264

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

An interval such as {3,4,5} is a set with all differences of adjacent elements equal to 1.

			The a(1) = 1 through a(3) = 9 set multipartitions (multisets of sets):
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,2}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Cf. A055887, A063834, A270995, A304969, A349050, A349055.
Intervals are counted by A000012, A001227, ranked by A073485.
Other conditions: A034691, A116540, A255906, A356933, A356942.
Other types: A107742, A356936, A356938, A356939.

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]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@allnorm[n],And@@chQ/@#&]],{n,0,5}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
R(n,k) = {EulerT(vector(n, j, max(0, 1+k-j)))}
seq(n) = {my(A=1+O(y*y^n)); for(k = 1, n, A += x^k*(1 + y*Ser(R(n,k), y) - polcoef(1/(1 - x*A) + O(x^(k+2)), k+1))); Vec(subst(A,x,1))} \\ Andrew Howroyd, Jan 01 2023

Terms a(10) and beyond from Andrew Howroyd, Jan 01 2023

A356938 Number of multisets of intervals whose multiset union is of size n and covers an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 7, 18, 41, 101, 228, 538, 1209

Offset: 0

Gus Wiseman, Sep 09 2022

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

			The a(1) = 1 through a(4) = 18 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{1}}  {{1},{1,2}}    {{1},{1,2,3}}
         {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}
                    {{3},{1,2}}    {{1},{2,3,4}}
                    {{1},{1},{1}}  {{1,2},{3,4}}
                    {{1},{1},{2}}  {{4},{1,2,3}}
                    {{1},{2},{3}}  {{1},{1},{1,2}}
                                   {{1},{1},{2,3}}
                                   {{1},{2},{1,2}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{1,2}}
                                   {{1},{4},{2,3}}
                                   {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{1},{2}}
                                   {{1},{1},{2},{2}}
                                   {{1},{1},{2},{3}}
                                   {{1},{2},{3},{4}}

Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
A011782 counts multisets covering an initial interval.
Intervals are counted by A000012, A001227, ranked by A073485.
Other conditions: A035310, A063834, A330783, A356934.
Other types: A107742, A356936, A356937, A356939, A356943, A356954.
Cf. A055887, A072233, A270995, A304969, A349050, A349055.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
Table[Length[Select[Join@@mps/@strnorm[n],And@@chQ/@#&]],{n,0,5}]

cp's OEIS Frontend

A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

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A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

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A307059 Expansion of 1/(2 - Product_{k>=1} (1 - x^k)).

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Magma

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

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Maple

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A371292 Numbers whose binary indices have prime indices covering an initial interval of positive integers.

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A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

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A336343 Number of ways to choose a strict partition of each part of a strict composition of n.

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