A356935 -id:A356935 - cp's OEIS frontend

A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

1, 1, 2, 4, 7, 13, 24, 42, 74, 130, 224, 383, 653, 1100, 1846, 3079, 5104, 8418, 13827, 22592, 36774, 59613, 96271, 154908, 248441, 397110, 632823, 1005445, 1592962, 2516905, 3966474, 6235107, 9777791, 15297678, 23880160, 37196958, 57819018, 89691934, 138862937

Offset: 0

Gus Wiseman, Sep 11 2022

nonn

			The a(1) = 1 through a(5) = 13 multiset partitions:
  {1}  {2}     {3}        {4}           {5}
       {1}{1}  {111}      {112}         {113}
               {1}{2}     {1}{3}        {122}
               {1}{1}{1}  {2}{2}        {1}{4}
                          {1}{111}      {2}{3}
                          {1}{1}{2}     {11111}
                          {1}{1}{1}{1}  {1}{112}
                                        {2}{111}
                                        {1}{1}{3}
                                        {1}{2}{2}
                                        {1}{1}{111}
                                        {1}{1}{1}{2}
                                        {1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Partitions with odd multiplicities are counted by A055922.
Odd-length multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other types: A050330, A356933, A356934, A356935.
Other conditions: A001970, A006171, A007294, A089259, A107742, A356941.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations.
Cf. A000670, A011782, A055887, A072233, A117958, A270995, A304969.

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],OddQ[Times@@Length/@#]&]],{n,0,8}]

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

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

Terms a(13) and beyond from Andrew Howroyd, Dec 30 2022

A358334 Number of twice-partitions of n into odd-length partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 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(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

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

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],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), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

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

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

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

Original entry on oeis.org

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[#]&]

A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
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 multiset partitions:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}

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

Gapless multisets are counted by A034296, ranked by A073491.
The initial version is A356955.
Other types: A356233, A356941, A356942, A356943.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
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.
A011782 counts multisets covering an initial interval.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A003963, A076610, A270995, A302242, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[100],And@@nogapQ/@primeMS/@primeMS[#]&]

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}]

A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 27, 28, 29, 31, 32, 33, 36, 38, 42, 44, 48, 49, 53, 54, 56, 57, 58, 59, 62, 63, 64, 66, 71, 72, 76, 77, 79, 81, 83, 84, 87, 88, 93, 96, 97, 98, 99, 106, 108, 112, 114, 116, 118, 121, 124, 126, 127

Offset: 1

Gus Wiseman, Sep 11 2022

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. 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 combined 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: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}

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

Multisets of odd numbers are counted by A000009, ranked by A066208.
Factorizations of this type are counted by A356931.
The version for odd lengths instead of parts is A356935, ranked by A089259.
Other conditions: A302478, A302492, A356939, A356940, 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. A000720, A003963, A011782, A055887, A076610, A302242, A304969.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@(OddQ[Times@@primeMS[#]]&/@primeMS[#])&]

A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

Original entry on oeis.org

1, 1, 2, 8, 28, 108, 524, 2608, 14176, 86576, 550672, 3782496, 27843880, 214071392, 1751823600, 15041687664, 134843207240, 1269731540864, 12427331494304, 126619822952928, 1341762163389920, 14712726577081248, 167209881188545344, 1963715680476759040, 23794190474350155856

Offset: 0

Gus Wiseman, Sep 08 2022

nonn

			The a(4) = 28 multiset partitions:
  {1}{111}      {1}{112}      {1}{123}      {1}{234}
  {1}{1}{1}{1}  {1}{122}      {1}{223}      {2}{134}
                {1}{222}      {1}{233}      {3}{124}
                {2}{111}      {2}{113}      {4}{123}
                {2}{112}      {2}{123}      {1}{2}{3}{4}
                {2}{122}      {2}{133}
                {1}{1}{1}{2}  {3}{112}
                {1}{1}{2}{2}  {3}{122}
                {1}{2}{2}{2}  {3}{123}
                              {1}{1}{2}{3}
                              {1}{2}{2}{3}
                              {1}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 0..500
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, A072233, A270995, A304969, A349050, A349055.
Odd-size multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other conditions: A034691, A116540, A255906, A356937, A356942.
Other types: A050330, A356932, A356934, A356935.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],OddQ[Times@@Length/@#]&]],{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, if(j%2 == 1, binomial(j+k-1, j))))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Jan 01 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2023

A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 96, 104, 108, 113, 117, 128, 144, 156, 162, 169, 192, 208, 216, 226, 234, 243, 256, 288, 312, 324, 338, 339, 351, 384, 416, 432, 452, 468, 486, 507, 512, 576, 624, 648

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
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: {{},{}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  32: {{},{},{},{},{}}
  36: {{},{},{1},{1}}
  39: {{1},{1,2}}
  48: {{},{},{},{},{1}}
  52: {{},{},{1,2}}
  54: {{},{1},{1},{1}}
  64: {{},{},{},{},{},{}}

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

This is the initial version of A356939.
Initial intervals are counted by A010054, ranked by A002110.
Other types: A007294, A322585.
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, A302242, A304969.

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

A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 26, 27, 28, 32, 36, 37, 38, 39, 42, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 72, 74, 76, 78, 81, 84, 89, 91, 96, 98, 104, 106, 108, 111, 112, 113, 114, 117, 122, 126, 128, 131, 133, 144, 147, 148, 151, 152

Offset: 1

Gus Wiseman, Sep 12 2022

nonn

An initial interval is a set {1,2,...,n}  for some n >= 0.
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: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  32: {{},{},{},{},{}}

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

Multisets covering an initial interval are ctd by A011782, rkd by A055932.
This is the initial version of A356944.
Other types: A034691, A089259, A356945, A356954.
Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
A000041 counts integer partitions, strict A000009.
A000670 counts patterns, ranked by A333217, necklace A019536.
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, A349050, A349055.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],And@@normQ/@primeMS/@primeMS[#]&]

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

cp's OEIS Frontend

A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

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A358334 Number of twice-partitions of n into odd-length partitions.

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

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A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

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

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A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

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A356933 Number of multisets of multisets, each of odd size, whose multiset union is a size-n multiset covering an initial interval.

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A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

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