A047968 -id:A047968 - cp's OEIS frontend

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

1, 1, 2, 4, 8, 17, 30, 61, 110, 207, 381, 711, 1250

Offset: 0

Gus Wiseman, Oct 11 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

			The a(1) = 1 through a(5) = 17 multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,1,1}}    {{2,2}}        {{1,1,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,2,2}}
                  {{1},{1,1}}  {{1},{3}}      {{1},{4}}
                               {{1,1,1,1}}    {{2},{3}}
                               {{1},{1,2}}    {{1,1,1,2}}
                               {{2},{1,1}}    {{1},{1,3}}
                               {{1},{1,1,1}}  {{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}}

Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

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]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 1, 3, 8, 27, 94, 385, 1673, 8079, 41614, 231447, 1364697, 8559575, 56544465, 393485452, 2867908008, 21869757215, 173848026202, 1438593095272, 12360614782433, 110119783919367, 1015289796603359, 9674959683612989, 95147388659652754, 964559157655032720, 10067421615492769230

Offset: 0

Gus Wiseman, Jan 02 2020

nonn

The (weakly) normal version is A116540.

			The a(1) = 1 through a(3) = 8 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{1}}  {{1},{1,2}}
         {{1},{2}}  {{1},{2,3}}
                    {{2},{1,3}}
                    {{3},{1,2}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{3}}
The a(4) = 27 set multipartitions:
  {{1},{1},{1},{1}}  {{1},{1},{1,2}}  {{1},{1,2,3}}  {{1,2,3,4}}
  {{1},{1},{1},{2}}  {{1},{1},{2,3}}  {{1,2},{1,2}}
  {{1},{1},{2},{2}}  {{1},{2},{1,2}}  {{1,2},{1,3}}
  {{1},{1},{2},{3}}  {{1},{2},{1,3}}  {{1},{2,3,4}}
  {{1},{2},{3},{4}}  {{1},{2},{3,4}}  {{1,2},{3,4}}
                     {{1},{3},{1,2}}  {{1,3},{2,4}}
                     {{1},{3},{2,4}}  {{1,4},{2,3}}
                     {{1},{4},{2,3}}  {{2},{1,3,4}}
                     {{2},{3},{1,4}}  {{3},{1,2,4}}
                     {{2},{4},{1,3}}  {{4},{1,2,3}}
                     {{3},{4},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Allowing edges to be multisets gives is A035310.
The strict case is A318402.
The constant case is A000005.
The (weakly) normal version is A116540.
Unlabeled set multipartitions are A049311.
Set multipartitions of prime indices are A050320.
Set multipartitions of integer partitions are A089259.
Cf. A001055, A047968, A255906, A269134, A283877, A296119, A317775, A318360, A318362, A330625, A330628.

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],And@@UnsameQ@@@#&]],{n,0,5}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n)))/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 8, 7, 19, 11, 24, 26, 38, 28, 85, 46, 89, 99, 146, 110, 246, 163, 326, 305, 416, 376, 816, 591, 903, 971, 1450, 1295, 2517, 1916, 3045, 3141, 4042, 4117, 7073, 5736, 8131, 9026, 12658, 11514, 19459, 16230, 24638, 27129, 33747, 32279, 55778, 45761, 71946

Offset: 0

Gus Wiseman, Oct 12 2018

nonn

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022

			The a(9) = 26 multiset partitions:
  {{9}}
  {{1,8}}
  {{2,7}}
  {{3,6}}
  {{4,5}}
  {{1,2,6}}
  {{1,3,5}}
  {{1},{8}}
  {{2,3,4}}
  {{2},{7}}
  {{3,3,3}}
  {{3},{6}}
  {{4},{5}}
  {{1},{2},{6}}
  {{1},{3},{5}}
  {{2},{3},{4}}
  {{3},{3},{3}}
  {{1,1,1,2,2,2}}
  {{1,1,1},{2,2,2}}
  {{1,1,2},{1,2,2}}
  {{1,1},{1,2},{2,2}}
  {{1,2},{1,2},{1,2}}
  {{1,1,1,1,1,1,1,1,1}}
  {{1,1,1},{1,1,1},{1,1,1}}
  {{1},{1},{1},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..150
A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.

Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.

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],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]

Terms a(11) and beyond from Andrew Howroyd, Feb 04 2022

A320698 Numbers whose product of prime indices is a prime power (A246655).

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 53, 54, 56, 57, 59, 62, 63, 67, 68, 72, 76, 80, 81, 82, 83, 84, 88, 92, 96, 97, 98, 100, 103, 106, 108, 109, 112, 114, 115, 118, 121, 124

Offset: 1

Gus Wiseman, Oct 19 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.

Also numbers whose prime indices are all powers of a common prime number.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (2), (3), (1,2), (4), (2,2), (1,3), (5), (1,1,2), (1,4), (7), (1,2,2), (8), (1,1,3), (2,4), (1,5), (9), (1,1,1,2), (3,3), (2,2,2), (1,1,4), (11), (1,7), (1,1,2,2), (1,8), (1,1,1,3), (13), (1,2,4), (1,1,5), (1,9), (1,1,1,1,2), (4,4), (1,3,3), (16), (1,2,2,2), (1,1,1,4), (2,8).

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A047968, A056239, A064573, A112798, A246547, A246655, A279787, A320322, A320325, A320699, A320700.

Select[Range[100],PrimePowerQ[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]]&]

is(n) = my(f=factor(n)[, 1]~, p=1); for(k=1, #f, p=p*primepi(f[k])); isprimepower(p) \\ Felix Fröhlich, Oct 20 2018

A323531 Number of square multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 9, 12, 18, 24, 36, 48, 69, 97, 139, 196, 283, 402, 576, 819, 1161, 1635, 2301, 3209, 4469, 6193, 8571, 11812, 16291, 22404, 30850, 42414, 58393, 80305, 110578, 152091, 209308, 287686, 395352, 542413, 743603, 1017489, 1390510, 1896482

Offset: 0

Gus Wiseman, Jan 21 2019

nonn

A multiset partition is square if the number of parts is equal to the number of parts in each part.

			The a(3) = 1 through a(9) = 12 square multiset partitions:
  (3)  (4)       (5)       (6)       (7)       (8)       (9)
       (11)(11)  (21)(11)  (21)(21)  (22)(21)  (22)(22)  (32)(22)
                           (22)(11)  (31)(21)  (31)(22)  (32)(31)
                           (31)(11)  (32)(11)  (31)(31)  (33)(21)
                                     (41)(11)  (32)(21)  (41)(22)
                                               (33)(11)  (41)(31)
                                               (41)(21)  (42)(21)
                                               (42)(11)  (43)(11)
                                               (51)(11)  (51)(21)
                                                         (52)(11)
                                                         (61)(11)
                                                         (111)(111)(111)

Cf. A000219, A001970, A047968, A261049, A279787, A305551, A319066, A323580.

Table[Sum[Length[Union@@(Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@#]]&/@IntegerPartitions[n,{k}])],{k,Sqrt[n]}],{n,30}]

A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

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, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Oct 31 2020

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.

Also Heinz numbers of partitions whose set of distinct parts is a singleton or pairwise coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
      2: {1}         17: {7}           33: {2,5}
      3: {2}         18: {1,2,2}       34: {1,7}
      4: {1,1}       19: {8}           35: {3,4}
      5: {3}         20: {1,1,3}       36: {1,1,2,2}
      6: {1,2}       22: {1,5}         37: {12}
      7: {4}         23: {9}           38: {1,8}
      8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
      9: {2,2}       25: {3,3}         41: {13}
     10: {1,3}       26: {1,6}         43: {14}
     11: {5}         27: {2,2,2}       44: {1,1,5}
     12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
     13: {6}         29: {10}          46: {1,9}
     14: {1,4}       30: {1,2,3}       47: {15}
     15: {2,3}       31: {11}          48: {1,1,1,1,2}

A302798 is the squarefree case.
A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.
A304711 does not consider singletons relatively prime, except for (1).
A304712 counts the partitions with these Heinz numbers.
A316476 is the version for indivisibility instead of relative primality.
A328867 is the pairwise non-coprime instead of pairwise coprime version.
A337600 counts triples of this type, with ordered version A337602.
A338330 is the complement.
A000961 lists powers of primes.
A051424 counts pairwise coprime or singleton partitions.
A304038 gives the distinct prime indices of each positive integer.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.

Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A304711 \/ A000961.

A168111 Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 8, 4, 10, 1, 22, 1, 18, 11, 30, 1, 47, 1, 57, 19, 59, 1, 121, 8, 104, 34, 158, 1, 242, 1, 261, 60, 300, 23, 514, 1, 493, 105, 706, 1, 959, 1, 1066, 217, 1258, 1, 1927, 16, 2010, 301, 2545, 1, 3442, 64, 3898, 494, 4568, 1, 6555, 1, 6845, 841, 8610

Offset: 1

Omar E. Pol, Nov 22 2009

Row sums of triangle A168021 except the first column.

Row sums of triangle A168016 except the last column.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000041, A001065, A047968, A168016, A168017, A168018, A168020, A168021.

A047968 := proc(n) add(combinat[numbpart](d), d= numtheory[divisors](n) ) ; end proc: A000041 := proc(n) combinat[numbpart](n) ; end proc: A168111 := proc(n) A047968(n)-A000041(n) ; end proc: seq(A168111(n),n=1..90) ; # R. J. Mathar, Jan 25 2010

a[ n_] := If[n < 1, 0, Sum[ PartitionsP[ d] Boole[ d < n], {d, Divisors @ n}]]; (* Michael Somos, Feb 24 2014 *)

A168111(n) = sumdiv(n,d,(dAntti Karttunen, Nov 14 2017

a(n) = A047968(n) - A000041(n).
G.f.: Sum_{n > 0} A000041(n)*x^(2*n)/(1-x^n). - Mircea Merca, Feb 24 2014
G.f.: x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 8*x^8 + 4*x^9 + 10*x^10 + x^11 + ... - Michael Somos, Feb 24 2014

Terms beyond a(12) from R. J. Mathar, Jan 25 2010

New name from Omar E. Pol, Feb 25 2014

A296134 Number of twice-factorizations of n of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of ways to choose a strict integer partition of a divisor of A052409(n).

			The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (4*4), (16).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsQ],{n,100}]

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A296134(n) = if(1==n,n,sumdiv(A052409(n),d,A000009(d))); \\ Antti Karttunen, Jul 29 2018

From Antti Karttunen, Jul 31 2018: (Start)
a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000009(d).
a(n) = A047966(A052409(n)). (End)

More terms from Antti Karttunen, Jul 29 2018

A323766 Dirichlet convolution of the integer partition numbers A000041 with the number of divisors function A000005.

Original entry on oeis.org

1, 1, 4, 5, 12, 9, 25, 17, 42, 39, 64, 58, 132, 103, 173, 200, 303, 299, 491, 492, 756, 832, 1122, 1257, 1858, 1975, 2646, 3083, 4057, 4567, 6118, 6844, 8913, 10265, 12912, 14931, 19089, 21639, 27003, 31397, 38830, 44585, 55138, 63263, 77371, 89585, 108076

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

Also the number of constant multiset partitions of constant multiset partitions of integer partitions of n.

			The a(6) = 25 constant multiset partitions of constant multiset partitions of integer partitions of 6:
  ((6))
  ((52))
  ((42))
  ((33))
  ((3)(3))
  ((3))((3))
  ((411))
  ((321))
  ((222))
  ((2)(2)(2))
  ((2))((2))((2))
  ((3111))
  ((2211))
  ((21)(21))
  ((21))((21))
  ((21111))
  ((111111))
  ((111)(111))
  ((11)(11)(11))
  ((111))((111))
  ((11))((11))((11))
  ((1)(1)(1)(1)(1)(1))
  ((1)(1)(1))((1)(1)(1))
  ((1)(1))((1)(1))((1)(1))
  ((1))((1))((1))((1))((1))((1))

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A000041, A000837, A001970, A034729, A047968, A306017, A319066, A323764, A323765, A323774.

Table[If[n==0,1,Sum[PartitionsP[d]*DivisorSigma[0,n/d],{d,Divisors[n]}]],{n,0,30}]

a(n) = if (n==0, 1, sumdiv(n, d, numbpart(d)*numdiv(n/d))); \\ Michel Marcus, Jan 28 2019

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 28 2019

A302596 Powers of prime numbers of prime index.

Original entry on oeis.org

1, 3, 5, 9, 11, 17, 25, 27, 31, 41, 59, 67, 81, 83, 109, 121, 125, 127, 157, 179, 191, 211, 241, 243, 277, 283, 289, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 625, 709, 729, 739, 773, 797, 859, 877, 919, 961, 967, 991, 1031, 1063, 1087, 1153

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set multisystems.
001: {}
003: {{1}}
005: {{2}}
009: {{1},{1}}
011: {{3}}
017: {{4}}
025: {{2},{2}}
027: {{1},{1},{1}}
031: {{5}}
041: {{6}}
059: {{7}}
067: {{8}}
081: {{1},{1},{1},{1}}
083: {{9}}
109: {{10}}
121: {{3},{3}}
125: {{2},{2},{2}}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

Select[Range[400],MatchQ[FactorInteger[#],{{?(#===1||PrimeQ[PrimePi[#]]&),}}]&]

Sum_{n>=1} 1/a(n) = 1 + Sum_{p in A006450} 1/(p-1). - Amiram Eldar, Sep 19 2022

cp's OEIS Frontend

A320331 Number of strict T_0 multiset partitions of integer partitions of n.

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A330783 Number of set multipartitions (multisets of sets) of strongly normal multisets of size n, where a finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

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A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

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A320698 Numbers whose product of prime indices is a prime power (A246655).

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A323531 Number of square multiset partitions of integer partitions of n.

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A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

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A168111 Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

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A296134 Number of twice-factorizations of n of type (R,Q,R).

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