A326521 -id:A326521 - cp's OEIS frontend

A326518 Number of normal multiset partitions of weight n where every part has the same sum.

1, 1, 3, 7, 15, 31, 75, 169, 445, 1199, 3471

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(4) = 15 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
             {{1},{1}}  {{1,2,2}}      {{1,1,2,2}}
                        {{1,2,3}}      {{1,1,2,3}}
                        {{2},{1,1}}    {{1,2,2,2}}
                        {{3},{1,2}}    {{1,2,2,3}}
                        {{1},{1},{1}}  {{1,2,3,3}}
                                       {{1,2,3,4}}
                                       {{1,1},{1,1}}
                                       {{1,2},{1,2}}
                                       {{1,3},{2,2}}
                                       {{1,4},{2,3}}
                                       {{2},{2},{1,1}}
                                       {{3},{3},{1,2}}
                                       {{1},{1},{1},{1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A035470, A038041, A255906, A317583, A321455, A326517, A326519, A326520, A326521, A326534.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Total/@#&]],{n,0,5}]

a(10) from Robert Price, Apr 04 2025

A326519 Number of normal multiset partitions of weight n where each part has a different sum.

Original entry on oeis.org

1, 1, 3, 11, 51, 259, 1461, 9133, 62348, 459547, 3632419

Offset: 0

Gus Wiseman, Jul 12 2019

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 11 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{2}}  {{1,2,2}}
                        {{1,2,3}}
                        {{1},{1,1}}
                        {{1},{1,2}}
                        {{1},{2,2}}
                        {{1},{2,3}}
                        {{2},{1,2}}
                        {{2},{1,3}}
                        {{1},{2},{3}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A275780, A317583, A321469, A326517, A326518, A326520, A326521, A326535.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Total/@#&]],{n,0,5}]

a(8)-a(10) from Robert Price, Apr 03 2025

A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

			The a(3) = 8 multiset partitions:
  {{1,1,1}}
  {{1,1,2}}
  {{1,2,2}}
  {{1,2,3}}
  {{1},{1},{1}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521, A331315.

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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Length/@#&]],{n,8}]

\\ here U(n,m) gives number for m blocks of size n.
U(n,m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )}
a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018

a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020

Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018

A326517 Number of normal multiset partitions of weight n where each part has a different size.

Original entry on oeis.org

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

A326513 Number of set partitions of {1..n} where each block has a different average.

Original entry on oeis.org

1, 1, 2, 4, 12, 40, 154, 650, 3024, 15110, 81538, 468494, 2863340, 18481390, 125838194, 897725927, 6715102246, 52372397021, 425716871241, 3594451206166, 31509992921241, 285799247349838, 2682935185643622, 25990339824995969, 259777696236210943, 2673388551328088666

Offset: 0

Gus Wiseman, Jul 11 2019

nonn

			The a(1) = 1 through a(4) = 12 set partitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
         {{1},{2}}  {{1},{2,3}}    {{1},{2,3,4}}
                    {{1,2},{3}}    {{1,2},{3,4}}
                    {{1},{2},{3}}  {{1,2,3},{4}}
                                   {{1,2,4},{3}}
                                   {{1,3},{2,4}}
                                   {{1,3,4},{2}}
                                   {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A275780, A326512, A326516, A326521, A326537.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],UnsameQ@@Mean/@#&]],{n,0,8}]

a(12) from Alois P. Heinz, Jul 12 2019

a(13)-a(25) from Christian Sievers, Aug 20 2024

A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 4, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 8, 2, 2, 2, 4, 1, 7, 2, 3, 2, 2, 2, 6, 1, 3, 3, 5, 1, 5, 1, 4, 4

Offset: 1

Gus Wiseman, Jul 12 2019

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.

			The a(60) = 8 factorizations: (2*5*6), (3*4*5), (2*30), (3*20), (4*15), (5*12), (6*10), (60).

Cf. A001055, A038041, A051293, A321455, A321469, A322794, A326513, A326514, A326515, A326521, A326537.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],UnsameQ@@Mean/@primeMS/@#&]],{n,100}]

avgpis(n) = { my(f=factor(n)); f[,1] = apply(primepi,f[,1]); (1/bigomega(n))*sum(i=1,#f~,f[i,2]*f[i,1]); };
all_have_different_average_of_pis(facs) = if(!#facs, 1, (#Set(apply(avgpis,facs)) == #facs));
A326516(n, m=n, facs=List([])) = if(1==n, all_have_different_average_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A326516(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A326537 MM-numbers of multiset partitions where each part has a different average.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

These are numbers where each prime index has a different average of prime indices. 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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of multiset partitions where each part has a different average, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A051293, A112798, A302242, A320324, A326513, A326516, A326521, A326533, A326534, A326535, A326536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Mean/@primeMS/@primeMS[#]&]

A326520 Number of normal multiset partitions of weight n where every part has the same average.

Original entry on oeis.org

1, 1, 3, 7, 17, 35, 103, 197

Offset: 0

Gus Wiseman, Jul 12 2019

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(4) = 17 normal multiset partitions where every part has the same average:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
             {{1},{1}}  {{1,2,2}}      {{1,1,2,2}}
                        {{1,2,3}}      {{1,1,2,3}}
                        {{1},{1,1}}    {{1,2,2,2}}
                        {{2},{1,3}}    {{1,2,2,3}}
                        {{1},{1},{1}}  {{1,2,3,3}}
                                       {{1,2,3,4}}
                                       {{1},{1,1,1}}
                                       {{1,1},{1,1}}
                                       {{1,2},{1,2}}
                                       {{1,3},{2,2}}
                                       {{1,4},{2,3}}
                                       {{2},{1,2,3}}
                                       {{1},{1},{1,1}}
                                       {{2},{2},{1,3}}
                                       {{1},{1},{1},{1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A051293, A255906, A317583, A326512, A326515, A326517, A326518, A326519, A326521, A326536.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Mean/@#&]],{n,0,5}]

A361866 Number of set partitions of {1..n} with block-means summing to an integer.

Original entry on oeis.org

1, 1, 1, 3, 8, 22, 75, 267, 1119, 4965, 22694, 117090, 670621, 3866503, 24113829, 161085223, 1120025702, 8121648620, 62083083115, 492273775141, 4074919882483

Offset: 0

Gus Wiseman, Apr 04 2023

			The a(1) = 1 through a(4) = 8 set partitions:
  {{1}}  {{1}{2}}  {{123}}      {{1}{234}}
                   {{13}{2}}    {{12}{34}}
                   {{1}{2}{3}}  {{123}{4}}
                                {{13}{24}}
                                {{14}{23}}
                                {{1}{24}{3}}
                                {{13}{2}{4}}
                                {{1}{2}{3}{4}}
The set partition y = {{1,2},{3,4}} has block-means {3/2,7/2}, with sum 5, so y is counted under a(4).

For mean instead of sum we have A361865, for median A361864.
For median instead of mean we have A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions with integer mean block-size.
A327475 counts subsets with integer mean, median A000975.
A327481 counts subsets by mean, median A013580.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513.
Cf. A067659, A326515, A326516, A326521.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Total[Mean/@#]]&]],{n,6}]

a(14)-a(20) from Christian Sievers, May 12 2025

A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

Original entry on oeis.org

1, 0, 3, 2, 12, 18, 101, 232, 1547, 3768, 24974, 116728, 687419, 3489664, 26436217, 159031250, 1129056772

Offset: 1

Gus Wiseman, Apr 04 2023

			The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}  {{12345}}
            {{13}{2}}    {{123}{4}}  {{1245}{3}}
            {{1}{2}{3}}              {{135}{24}}
                                     {{15}{234}}
                                     {{1}{234}{5}}
                                     {{12}{3}{45}}
                                     {{135}{2}{4}}
                                     {{14}{25}{3}}
                                     {{15}{24}{3}}
                                     {{1}{24}{3}{5}}
                                     {{15}{2}{3}{4}}
                                     {{1}{2}{3}{4}{5}}

For median instead of mean we have A361864.
For sum instead of outer mean we have A361866, median A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions whose block-sizes have integer mean.
A327475 counts subsets with integer mean, median A000975.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]

a(13)-a(17) from Christian Sievers, Jun 30 2025

cp's OEIS Frontend

A326518 Number of normal multiset partitions of weight n where every part has the same sum.

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A326519 Number of normal multiset partitions of weight n where each part has a different sum.

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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

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A326517 Number of normal multiset partitions of weight n where each part has a different size.

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A326513 Number of set partitions of {1..n} where each block has a different average.

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A326516 Number of factorizations of n into factors > 1 where each factor has a different average of prime indices.

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A326537 MM-numbers of multiset partitions where each part has a different average.

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