A330459 -id:A330459 - 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

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).

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.

A330458 Number of multisets of nonempty sets of nonempty multisets of positive integers with total sum n.

Original entry on oeis.org

1, 1, 3, 8, 20, 49, 123, 292, 701, 1653, 3874, 8977, 20711, 47344, 107692, 243382, 547264, 1224048, 2725483, 6040796, 13334354, 29316445, 64215841, 140159357, 304890958, 661097630, 1429083295, 3080159882, 6620188725, 14190463947, 30338920339, 64702805452

Offset: 0

Gus Wiseman, Dec 17 2019

nonn

			The a(4) = 20 partitions:
  ((4))  ((22))      ((13))      ((112))          ((1111))
         ((2))((2))  ((1)(3))    ((1)(12))        ((1)(111))
                     ((1))((3))  ((2)(11))        ((1))((111))
                                 ((1))((12))      ((11))((11))
                                 ((2))((11))      ((1))((1)(11))
                                 ((1))((1)(2))    ((1))((1))((11))
                                 ((1))((1))((2))  ((1))((1))((1))((1))

Cf. A001970, A007713, A050342, A050343, A063834, A089259, A261049, A270995, A271619, 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@@@#&]],{n,0,10}]

Euler transform of A261049. 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.

A330464 Number of non-isomorphic weight-n sets of set-systems with distinct multiset unions.

Original entry on oeis.org

1, 1, 3, 9, 32, 111, 463, 1942

Offset: 0

Gus Wiseman, Dec 26 2019

A set-system is a finite set of finite nonempty sets of positive integers.

As an alternative description, a(n) is the number of non-isomorphic sets of sets of sets with n leaves where the inner sets of sets all have different multiset unions.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets:
  {}  {{{1}}}  {{{1,2}}}      {{{1,2,3}}}
               {{{1},{2}}}    {{{1},{1,2}}}
               {{{1}},{{2}}}  {{{1},{2,3}}}
                              {{{1}},{{1,2}}}
                              {{{1}},{{2,3}}}
                              {{{1},{2},{3}}}
                              {{{1}},{{1},{2}}}
                              {{{1}},{{2},{3}}}
                              {{{1}},{{2}},{{3}}}

Non-isomorphic sets of sets are A283877.
Non-isomorphic sets of sets of sets are A323790.
Non-isomorphic set partitions of set-systems are A323795.
Cf. A089259, A141268, A271619, A279375, A279785, A306186, A316980, A317533, A318564, A318565, A318566, A330459, A330472.

cp's OEIS Frontend

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

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

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A330458 Number of multisets of nonempty sets of nonempty multisets of positive integers with total sum n.

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