A261049 -id:A261049 - cp's OEIS frontend

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

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.

A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 3, 6, 7, 14, 15, 32, 31, 63, 56, 142, 101, 240, 211, 467, 297, 985, 490, 1524, 1247, 2542, 1255, 6371, 1979, 7486, 7070, 14128, 4565, 32953, 6842, 42229, 37863, 56266, 17887, 192914, 21637, 145820, 197835, 371853, 44583, 772740, 63261, 943966, 1124840

Offset: 0

Gus Wiseman, Dec 14 2022

nonn

			The a(1) = 1 through a(6) = 14 sets:
  {(1)}  {(2)}   {(3)}    {(4)}       {(5)}      {(6)}
         {(11)}  {(21)}   {(22)}      {(32)}     {(33)}
                 {(111)}  {(31)}      {(41)}     {(42)}
                          {(211)}     {(221)}    {(51)}
                          {(1111)}    {(311)}    {(222)}
                          {(2),(11)}  {(2111)}   {(321)}
                                      {(11111)}  {(411)}
                                                 {(2211)}
                                                 {(3111)}
                                                 {(21111)}
                                                 {(111111)}
                                                 {(3),(21)}
                                                 {(3),(111)}
                                                 {(21),(111)}

This is the constant-sum case of A261049, ordered A358906.
The version for all different sums is A271619, ordered A336342.
Allowing repetition gives A305551, ordered A279787.
The version for compositions instead of partitions is A358904.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A098407 counts sets of compositions, ordered A358907.
Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.

Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]

a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ Michel Marcus, Dec 14 2022

a(n) = Sum_{d|n} binomial(A000041(d),n/d).

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 70, 123, 208, 355, 597

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(6) = 23 antichains:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                  {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1},{3}}    {{1,2,2}}      {{1,1,4}}
                             {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
                             {{2},{1,1}}  {{2},{3}}      {{1},{5}}
                                          {{1,1,1,2}}    {{2,2,2}}
                                          {{1},{2,2}}    {{2},{4}}
                                          {{3},{1,1}}    {{1,1,1,3}}
                                          {{1,1,1,1,1}}  {{1,1,2,2}}
                                          {{1,1},{1,2}}  {{1},{2,3}}
                                          {{2},{1,1,1}}  {{2},{1,3}}
                                                         {{3},{1,2}}
                                                         {{4},{1,1}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1},{2,2}}
                                                         {{1},{2},{3}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A258466, A261049, A319719, A319721, A320353, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]

A320450 Number of strict antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 10, 13, 19, 28, 47, 64, 98

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(8) = 19 antichains:
  {{1}}  {{2}}  {{3}}      {{4}}      {{5}}      {{6}}
                {{1,2}}    {{1,3}}    {{1,4}}    {{1,5}}
                {{1},{2}}  {{1},{3}}  {{2,3}}    {{2,4}}
                                      {{1},{4}}  {{1,2,3}}
                                      {{2},{3}}  {{1},{5}}
                                                 {{2},{4}}
                                                 {{1},{2,3}}
                                                 {{2},{1,3}}
                                                 {{3},{1,2}}
                                                 {{1},{2},{3}}
.
  {{7}}          {{8}}
  {{1,6}}        {{1,7}}
  {{2,5}}        {{2,6}}
  {{3,4}}        {{3,5}}
  {{1,2,4}}      {{1,2,5}}
  {{1},{6}}      {{1,3,4}}
  {{2},{5}}      {{1},{7}}
  {{3},{4}}      {{2},{6}}
  {{1},{2,4}}    {{3},{5}}
  {{2},{1,4}}    {{1},{2,5}}
  {{4},{1,2}}    {{1},{3,4}}
  {{1,2},{1,3}}  {{2},{1,5}}
  {{1},{2},{4}}  {{3},{1,4}}
                 {{4},{1,3}}
                 {{5},{1,2}}
                 {{1,2},{1,4}}
                 {{1,2},{2,3}}
                 {{1},{2},{5}}
                 {{1},{3},{4}}

Cf. A001970, A050342, A089259, A258466, A261049, A319719, A319721, A320328, A320331, A320353.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

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.

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.

cp's OEIS Frontend

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

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A359041 Number of finite sets of integer partitions with all equal sums and total sum n.

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A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

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A320450 Number of strict antichains of sets whose multiset union is an integer partition 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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A330455 Number of sets 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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