A050343 -id:A050343 - cp's OEIS frontend

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

1, 1, 2, 7, 13, 34, 81, 175, 403, 890, 1977, 4262, 9356, 19963, 42573, 90865, 191206, 401803, 837898, 1744231, 3607504, 7436628, 15254309, 31185686, 63552725, 128963236, 260933000, 526140540, 1057927323, 2120500885, 4239012067, 8449746787, 16799938614

Offset: 0

Gus Wiseman, Dec 16 2019

nonn

Number of sets of disjoint nonempty sets of nonempty multisets of positive integers with total sum n.

			The a(4) = 13 partitions:
  ((4))  ((22))  ((31))      ((211))      ((1111))
                 ((1)(3))    ((1)(21))    ((1)(111))
                 ((1))((3))  ((2)(11))    ((1))((111))
                             ((1))((21))
                             ((2))((11))

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

Cf. A001970, A007713, A050343, A063834, A089259, A261049, A271619, A279375, A294617, A318565, A323787-A323795, A330452-A330459, A330460.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],UnsameQ@@Join@@#&]],{n,0,10}]

\\ here BellP is A000110 as series.
BellP(n)={serlaplace(exp( exp(x + O(x*x^n)) - 1))}
seq(n)={my(b=BellP(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^numbpart(k)))); vector(#v, n, my(r=v[n]); sum(k=0, n-1, polcoeff(b,k)*polcoef(r,k)))} \\ Andrew Howroyd, Dec 29 2019

a(n) = Sum_{0 <= k <= n} A330463(n,k) * A000110(k).

Terms a(18) and beyond from Andrew Howroyd, Dec 29 2019

A330461 Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 4, 4, 1, 1, 1, 1, 4, 7, 7, 5, 1, 1, 1, 1, 5, 12, 14, 11, 6, 1, 1, 1, 1, 6, 19, 29, 25, 16, 7, 1, 1, 1, 1, 8, 30, 57, 60, 41, 22, 8, 1, 1, 1, 1, 10, 49, 110, 141, 111, 63, 29, 9, 1, 1, 1

Offset: 0

Gus Wiseman, Dec 18 2019

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6
      -----------------------------
  n=0:  1   1   1   1   1   1   1
  n=1:  1   1   1   1   1   1   1
  n=2:  1   1   1   1   1   1   1
  n=3:  1   2   3   4   5   6   7
  n=4:  1   2   4   7  11  16  22
  n=5:  1   3   7  14  25  41  63
  n=6:  1   4  12  29  60 111 189
For example, the A(5,3) = 14 partitions are:
  {{5}}      {{1}}{{4}}
  {{14}}     {{2}}{{3}}
  {{23}}     {{1}}{{13}}
  {{1}{4}}   {{2}}{{12}}
  {{2}{3}}   {{1}}{{1}{3}}
  {{1}{13}}  {{2}}{{1}{2}}
  {{2}{12}}  {{1}}{{1}{12}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns are A000012 (k = 0), A000009 (k = 1), A050342 (k = 2), A050343 (k = 3), A050344 (k = 4).
The non-strict version is A290353.
Cf. A001970, A004111, A007713, A060016, A273873, A279375, A279785, A294617, A306186, A323718, A323790, A330462.

spl[n_,0]:={n};
spl[n_,k_]:=Select[Join@@Table[Union[Sort/@Tuples[spl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}],UnsameQ@@#&];
Table[Length[spl[n-k,k]],{n,0,10},{k,0,n}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
M(n, k=n)={my(L=List(), v=vector(n,i,1)); listput(L, concat([1], v)); for(j=1, k, v=WeighT(v); listput(L, concat([1], v))); Mat(Col(L))~}
{ my(A=M(7)); for(i=1, #A, print(A[i,])) } \\ Andrew Howroyd, Dec 31 2019

Column k is the k-th weigh transform of the all-ones sequence. The weigh transform of a sequence b has generating function Product_{i > 0} (1 + x^i)^b(i).

A050344 Number of partitions of n into distinct parts with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 11, 25, 60, 141, 321, 742, 1688, 3810, 8580, 19225, 42844, 95156, 210480, 463866, 1018957, 2231114, 4870400, 10601805, 23015117, 49833471, 107636878, 231940988, 498671281, 1069826434, 2290402343, 4893782240, 10436263572, 22214850439, 47202869437

Offset: 0

Christian G. Bower, Oct 15 1999

nonn

			4 = (((4))) = (((3)))+(((1))) = (((3))+((1))) = ((3)+(1)) = ((3+1)) = ((2+1))+((1)) = ((2+1)+(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A050342-A050350.

g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      g(n, i-1)+`if`(i>n, 0, g(n-i, i-1))))
    end:
h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(g(i, i), j)*h(n-i*j, i-1), j=0..n/i)))
    end:
f:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(h(i, i), j)*f(n-i*j, i-1), j=0..n/i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(f(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 19 2013

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, g[n, i - 1] + If[i > n, 0, g[n - i, i - 1]]]];
h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i, i], j]* h[n - i*j, i - 1], {j, 0, n/i}]]];
f[n_, i_] := f[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[h[i, i], j]* f[n - i*j, i - 1], {j, 0, n/i}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[f[i, i], j]* b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Weigh transform of A050343.

A050347 Number of ways to factor n into distinct factors with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 4, 1, 4, 1, 10, 1, 4, 4, 7, 1, 10, 1, 10, 4, 4, 1, 26, 1, 4, 4, 10, 1, 22, 1, 14, 4, 4, 4, 34, 1, 4, 4, 26, 1, 22, 1, 10, 10, 4, 1, 63, 1, 10, 4, 10, 1, 26, 4, 26, 4, 4, 1, 74, 1, 4, 10, 29, 4, 22, 1, 10, 4, 22, 1, 105, 1, 4, 10, 10, 4, 22, 1, 63, 7, 4, 1, 74, 4, 4, 4, 26

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).

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

R. J. Mathar, Table of n, a(n) for n = 1..2159

Cf. A045778, A050345-A050350. a(p^k)=A050343. a(A002110)=A000307.

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A050345(n).

a(n) = A050348(A101296(n)). - R. J. Mathar, May 26 2017

A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

Original entry on oeis.org

1, 1, 3, 5, 9, 10, 22, 20, 37, 44, 65, 68, 127, 119, 182, 226, 307, 335, 511, 544, 782, 913, 1171, 1359, 1908, 2121, 2738, 3286, 4174, 4821, 6305, 7182, 9108, 10739, 13195, 15548, 19465, 22397, 27477, 32423, 39448, 45843, 55995, 64871, 78343, 91761, 109325

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

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

			The a(1) = 1 through a(5) = 10 strict multiset partitions of constant multiset partitions of integer partitions:
  ((1))  ((2))     ((3))          ((4))             ((5))
         ((11))    ((21))         ((31))            ((41))
         ((1)(1))  ((111))        ((22))            ((32))
                   ((1)(1)(1))    ((211))           ((311))
                   ((1))((1)(1))  ((1111))          ((221))
                                  ((2)(2))          ((2111))
                                  ((11)(11))        ((11111))
                                  ((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. A000009, A000041, A001970, A034729, A047968, A050343, A316980, A319066, A323764, A323766, A323774.

Join[{1}, Table[Sum[PartitionsQ[d]*PartitionsP[n/d],{d,Divisors[n]}],{n,1,100}]]

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

A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 9, 23, 62, 161, 410, 1031, 2579, 6359, 15575, 37830, 91241, 218581, 520544, 1232431, 2902644, 6802178, 15866054, 36844016, 85202436, 196251933, 450341874, 1029709478, 2346409350, 5329371142, 12066816905, 27240224766, 61317231288, 137643961196

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

Number of sets of nonempty multisets of nonempty multisets of positive integers with total sum n.

			The a(4) = 23 partitions:
  ((4))  ((22))    ((31))      ((211))        ((1111))
         ((2)(2))  ((1)(3))    ((1)(21))      ((1)(111))
                   ((1))((3))  ((2)(11))      ((11)(11))
                               ((1)(1)(2))    ((1))((111))
                               ((1))((21))    ((1)(1)(11))
                               ((2))((11))    ((1))((1)(11))
                               ((1))((1)(2))  ((1)(1)(1)(1))
                               ((2))((1)(1))  ((11))((1)(1))
                                              ((1))((1)(1)(1))

Alois P. Heinz, Table of n, a(n) for n = 0..3853

The not necessarily strict case is A007713.

Cf. A001055, A001970, A050336, A050343, A089259, A261049, A271619, A316980, A318566, A323787-A323795, A330452-A330459, A330461, A330463.

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) a(n):= `if`(n<2, 1, add(a(n-k)*add(b(d)
      *d*(-1)^(k/d+1), d=divisors(k)), k=1..n)/n)
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jul 18 2021

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],UnsameQ@@#&]],{n,0,10}]

Weigh transform of A001970. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A330456 Number of multisets of nonempty sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 43, 84, 168, 332, 650, 1255, 2428, 4636, 8827, 16702, 31457, 58919, 109977, 204286, 378135, 697240, 1281315, 2346612, 4284654, 7799248, 14157079, 25626996, 46269838, 83330373, 149717844, 268371413, 479996794, 856661792, 1525761119, 2712050472

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

			The a(4) = 10 partitions:
  ((4))  ((13))      ((1)(12))        ((2))((2))  ((1))((1))((1))((1))
         ((1)(3))    ((1))((12))
         ((1))((3))  ((1))((1)(2))
                     ((1))((1))((2))

Cf. A001055, A001970, A007713, A050342, A050343, A063834, A089259, A270995, A279785, A330461, A323787-A323795, A330452-A330459.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[And@@UnsameQ@@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

Euler transform of A050342. The Euler transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} 1/(1 - x^i)^s_i.

A330454 Number of sets of nonempty sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 7, 15, 39, 94, 224, 526, 1236, 2857, 6568, 15003, 34030, 76757, 172216, 384386, 853960, 1888891, 4160524, 9128355, 19953661, 43463021, 94354292, 204182435, 440505489, 947590424, 2032730905, 4348897216, 9280361316, 19755155955, 41953293592, 88891338202

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

			The a(4) = 15 partitions:
  ((4))  ((22))  ((13))      ((112))        ((1111))
                 ((1)(3))    ((1)(12))      ((1)(111))
                 ((1))((3))  ((2)(11))      ((1))((111))
                             ((1))((12))    ((1))((1)(11))
                             ((2))((11))
                             ((1))((1)(2))

Cf. A001970, A007713, A050336, A050342, A050343, A261049, A271619, A279785, A330461, A330463, A323787-A323795, A330452-A330459.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[UnsameQ@@#,And@@UnsameQ@@@#]&]],{n,0,10}]

Weigh transform of A261049. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A330455 Number of sets of nonempty multisets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 2, 6, 12, 28, 62, 134, 285, 610, 1277, 2661, 5506, 11305, 23064, 46803, 94406, 189484, 378522, 752668, 1490319, 2939093, 5774065, 11302564, 22048496, 42869613, 83091843, 160569590, 309398958, 594532990, 1139416396, 2178119059, 4153507514, 7901706341

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

			The a(4) = 12 partitions:
  ((4))  ((2)(2))  ((13))      ((1)(12))      ((1)(1)(1)(1))
                   ((1)(3))    ((1)(1)(2))    ((1))((1)(1)(1))
                   ((1))((3))  ((1))((12))
                               ((1))((1)(2))
                               ((2))((1)(1))

Cf. A001055, A001970, A007713, A050336, A050343, A089259, A261049, A270995, A271619, A323787-A323795, A330452-A330459, A330461.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And[UnsameQ@@#,And@@UnsameQ@@@Join@@#]&]],{n,0,10}]

Weigh transform of A089259. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

A330457 Number of multisets of nonempty multisets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 3, 7, 17, 37, 87, 187, 414, 887, 1903, 4008, 8437, 17519, 36255, 74384, 151898, 308129, 622269, 1249768, 2499392, 4975421, 9865122, 19481300, 38331536, 75149380, 146840801, 285990797, 555297342, 1074996017, 2075201544, 3995079507, 7671034324, 14692086594

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

			The a(4) = 17 partitions:
  ((4))  ((13))      ((1)(12))        ((2)(2))    ((1)(1)(1)(1))
         ((1)(3))    ((1)(1)(2))      ((2))((2))  ((1))((1)(1)(1))
         ((1))((3))  ((1))((12))                  ((1)(1))((1)(1))
                     ((1))((1)(2))                ((1))((1))((1)(1))
                     ((2))((1)(1))                ((1))((1))((1))((1))
                     ((1))((1))((2))

Cf. A001970, A007713, A049311, A050343, A063834, A089259, A270995, A318564, A323787-A323795, A330452-A330459.

ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];
Table[Length[Select[ppl[n,3],And@@UnsameQ@@@Join@@#&]],{n,0,10}]

Euler transform of A089259. The Euler transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} 1/(1 - x^i)^s_i.

cp's OEIS Frontend

A330452 Number of set partitions of strict multiset partitions of integer partitions of n.

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A330461 Array read by antidiagonals where A(n,k) is the number of multiset partitions with k levels that are strict at all levels and have total sum n.

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A050344 Number of partitions of n into distinct parts with 3 levels of parentheses.

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A050347 Number of ways to factor n into distinct factors with 2 levels of parentheses.

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A323765 Dirichlet convolution of the integer partition numbers A000041 with the strict partition numbers A000009.

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A330453 Number of strict multiset partitions of multiset partitions of integer partitions of n.

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A330456 Number of multisets of nonempty sets of nonempty sets of positive integers with total sum n.

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A330454 Number of sets of nonempty sets of nonempty multisets of positive integers with total sum n.

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