A330463 -id:A330463 - cp's OEIS frontend

A358906 Number of finite sequences of distinct integer partitions with total sum n.

1, 1, 2, 7, 13, 35, 87, 191, 470, 1080, 2532, 5778, 13569, 30715, 69583, 160386, 360709, 814597, 1824055, 4102430, 9158405, 20378692, 45215496, 100055269, 221388993, 486872610, 1069846372, 2343798452, 5127889666, 11186214519, 24351106180, 52896439646

Offset: 0

Gus Wiseman, Dec 07 2022

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

This is the case of A055887 with distinct partitions.
The unordered version is A261049.
The case of twice-partitions is A296122.
The case of distinct sums is A336342, constant sums A279787.
The version for sequences of compositions is A358907.
The case of weakly decreasing lengths is A358908.
The case of distinct lengths is A358912.
The version for strict partitions is A358913, distinct case of A304969.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions.
A358830 counts twice-partitions with distinct lengths.
A358901 counts partitions with all distinct Omegas.
Cf. A000009, A000041, A000219, A098407, A271619, A330463, A358836, A358901, A358914.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, Feb 13 2024

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

a(n) = Sum_{k} A330463(n,k) * k!.

A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 2, 0, 0, 0, 0, 5, 11, 3, 0, 0, 0, 0, 0, 6, 16, 8, 0, 0, 0, 0, 0, 0, 8, 25, 15, 1, 0, 0, 0, 0, 0, 0, 10, 35, 28, 4, 0, 0, 0, 0, 0, 0, 0, 12, 52, 46, 9, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  2  0  0
  0  3  4  0  0  0
  0  4  6  2  0  0  0
  0  5 11  3  0  0  0  0
  0  6 16  8  0  0  0  0  0
  0  8 25 15  1  0  0  0  0  0
  0 10 35 28  4  0  0  0  0  0  0
  ...
Row n = 7 counts the following set-systems:
  {{7}}      {{1},{6}}      {{1},{2},{4}}
  {{1,6}}    {{2},{5}}      {{1},{2},{1,3}}
  {{2,5}}    {{3},{4}}      {{1},{3},{1,2}}
  {{3,4}}    {{1},{1,5}}
  {{1,2,4}}  {{1},{2,4}}
             {{2},{1,4}}
             {{2},{2,3}}
             {{3},{1,3}}
             {{4},{1,2}}
             {{1},{1,2,3}}
             {{1,2},{1,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)

Row sums are A050342.
Column k = 1 is A000009.
Cf. A001970, A050343, A063834, A270995, A271619, A279375, A279785, A283877, A294617, A326031, A330456, A330460, A330463, A360764.

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,2],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[#]==k]&]],{n,0,10},{k,0,n}]

L(n)={eta(x^2 + O(x*x^n))/eta(x + O(x*x^n))}
A(n)={my(c=L(n), v=Vec(prod(k=1, n, (1 + x^k*y + O(x*x^n))^polcoef(c,k)))); vector(#v, n, Vecrev(v[n],n))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

G.f.: Product_{j>=1} (1 + y*x^j)^A000009(j). - Andrew Howroyd, Dec 29 2019

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

Original entry on oeis.org

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

A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 0, 0, 0, 0, 4, 5, 1, 0, 0, 0, 0, 5, 6, 1, 0, 0, 0, 0, 0, 6, 9, 2, 0, 0, 0, 0, 0, 0, 8, 13, 3, 0, 0, 0, 0, 0, 0, 0, 10, 23, 10, 1, 0, 0, 0, 0, 0, 0, 0, 12, 27, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 15, 40, 19, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Dec 18 2019

			Triangle begins:
  1
  0  1
  0  1  0
  0  2  1  0
  0  2  1  0  0
  0  3  2  0  0  0
  0  4  5  1  0  0  0
  0  5  6  1  0  0  0  0
  0  6  9  2  0  0  0  0  0
  0  8 13  3  0  0  0  0  0  0
  0 10 23 10  1  0  0  0  0  0  0
  0 12 27 11  1  0  0  0  0  0  0  0
  0 15 40 19  2  0  0  0  0  0  0  0  0
Row n = 8 counts the following set partitions:
  {{8}}      {{1},{7}}    {{1},{2},{5}}
  {{3,5}}    {{2},{6}}    {{1},{3},{4}}
  {{2,6}}    {{3},{5}}
  {{1,7}}    {{1},{3,4}}
  {{1,3,4}}  {{1},{2,5}}
  {{1,2,5}}  {{2},{1,5}}
             {{3},{1,4}}
             {{4},{1,3}}
             {{5},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n=0..50)

Row sums are A294617.
Column k = 1 is A000009 (n > 0).
Cf. A000110, A008277, A050342, A060016, A072706, A270995, A271619, A279375, A279785, A326701, A330459, A330462, A330463, A330759.

b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 k*
         b(n-i, t, k)+b(n-i, t, k+1))(min(n-i, i-1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, Dec 29 2019

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,2],Length[#]==k&&And[UnsameQ@@#,UnsameQ@@Join@@#]&]],{n,0,10},{k,0,n}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + Function[t, k*b[n-i, t, k] + b[n-i, t, k + 1]][Min[n-i, i-1]]]];
T[n_] := PadRight[CoefficientList[b[n, n, 0], x], n + 1];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

A(n)={my(v=Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n)))); vector(#v, n, my(p=v[n]); vector(n, k, sum(i=k, n, polcoef(p,i-1)*stirling(i-1, k-1, 2))))}
{my(T=A(12)); for(n=1, #T, print(T[n]))} \\ Andrew Howroyd, Dec 29 2019

T(n,k) = Sum_{k <= i <= n} A060016(n,i) * A008277(i,k).

For n > 0, T(n,2) = Sum_{k = 1..n} (2^(k - 1) -1) * A060016(n,k).

A360742 Number T(n,k) of sets of nonempty integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 2, 0, 1, 4, 6, 5, 3, 0, 1, 5, 10, 10, 7, 4, 0, 1, 6, 14, 19, 16, 10, 5, 0, 1, 7, 19, 30, 32, 24, 14, 6, 0, 1, 8, 26, 46, 57, 52, 35, 19, 8, 0, 1, 9, 32, 67, 94, 97, 79, 50, 25, 10, 0, 1, 10, 40, 93, 147, 172, 157, 117, 69, 33, 12

Offset: 0

Alois P. Heinz, Feb 18 2023

			T(6,3) = 10: {[1,1,4]}, {[1,2,3]}, {[2,2,2]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[2],[2,2]}, {[3],[1,2]}, {[4],[1,1]}, {[1],[2],[3]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2,  2;
  0, 1, 3,  3,  2;
  0, 1, 4,  6,  5,  3;
  0, 1, 5, 10, 10,  7,  4;
  0, 1, 6, 14, 19, 16, 10,  5;
  0, 1, 7, 19, 30, 32, 24, 14,  6;
  0, 1, 8, 26, 46, 57, 52, 35, 19,  8;
  0, 1, 9, 32, 67, 94, 97, 79, 50, 25, 10;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-2 give: A000007, A057427, A001477(n-1) for n>=1.
Main diagonal gives A000009.
T(n+2,n+1) gives A036469.
Row sums give A261049.
T(2n,n) gives A360714.
Cf. A000041, A055884 (similar triangle for multisets), A330463.

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i), k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..12);

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[       g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i], k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
T[n_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 15 2023, after Alois P. Heinz *)

T(n,n) + T(n+1,n) = T(n+2,n+1) for n>=0.

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.

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.

cp's OEIS Frontend

A358906 Number of finite sequences of distinct integer partitions with total sum n.

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A330462 Triangle read by rows where T(n,k) is the number of k-element sets of nonempty sets of positive integers with total sum n.

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

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A330460 Triangle read by rows where T(n,k) is the number of set partitions with k blocks and total sum n.

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A360742 Number T(n,k) of sets of nonempty integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A330453 Number of strict multiset partitions of multiset partitions of integer partitions of 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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