A255906 -id:A255906 - cp's OEIS frontend

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691

Offset: 1

Gus Wiseman, Oct 13 2018

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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices 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 terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   667: {{2,2},{1,3}}
  1261: {{3,3},{1,2}}
  1363: {{1,3},{2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2093: {{1,1},{2,2},{1,2}}
  2117: {{1,3},{2,4}}
  2639: {{1,1},{1,2},{1,3}}
  4277: {{1,1},{1,2},{2,3}}
  4669: {{1,1},{2,2},{1,3}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

Original entry on oeis.org

1, 2, 2, 4, 8, 4, 8, 34, 26, 8, 16, 124, 168, 76, 16, 32, 448, 962, 674, 208, 32, 64, 1568, 5224, 5344, 2392, 544, 64, 128, 5448, 27336, 39834, 24578, 7816, 1376, 128, 256, 18768, 139712, 283864, 236192, 99832, 24048, 3392, 256, 512, 64448, 702496, 1960320, 2161602, 1186866, 370976, 70656, 8192, 512

Offset: 1

Gus Wiseman, Jul 30 2018

			The T(3,2) = 8 multiset partitions:
  {{1},{1,1}}
  {{1},{2,2}}
  {{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
Triangle begins:
    1
    2    2
    4    8    4
    8   34   26    8
   16  124  168   76   16
   32  448  962  674  208   32
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums are A255906.

Cf. A007716, A034691, A255397, A255903.

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],Length[#]==k&]],{n,7},{k,n}]

\\ here B(n,k) is A239473(n,k).
B(n,k)={sum(r=k, n, binomial(r, k)*(-1)^(r-k))}
Row(n)={Vecrev(sum(j=1, n, B(n,j)*polcoef(1/prod(k=1, n, (1 - x^k*y + O(x*x^n))^binomial(k+j-1,j-1)), n))/y)}
{ for(n=1, 10, print(Row(n))) } \\ Andrew Howroyd, Dec 31 2019

Terms a(29) and beyond from Andrew Howroyd, Dec 31 2019

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Original entry on oeis.org

1, 6, 21, 104, 452, 2335, 11992, 66810, 385101, 2336352, 14738380, 96831730, 659809115, 4657075074, 33974259046, 255781455848, 1984239830571, 15839628564349, 129951186405574, 1094486382191624, 9453318070371926, 83654146992936350, 757769011659766015, 7020652591448497490

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

			Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
  {{{1,1,1}}}
  {{{1,1,2}}}
  {{{1,2,3}}}
  {{{1},{1,1}}}
  {{{1},{1,2}}}
  {{{1},{2,3}}}
  {{{2},{1,1}}}
  {{{1},{1},{1}}}
  {{{1},{1},{2}}}
  {{{1},{2},{3}}}
  {{{1}},{{1,1}}}
  {{{1}},{{1,2}}}
  {{{1}},{{2,3}}}
  {{{2}},{{1,1}}}
  {{{1}},{{1},{1}}}
  {{{1}},{{1},{2}}}
  {{{1}},{{2},{3}}}
  {{{2}},{{1},{1}}}
  {{{1}},{{1}},{{1}}}
  {{{1}},{{1}},{{2}}}
  {{{1}},{{2}},{{3}}}

Cf. A001970, A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318564, A318565.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
dubnorm[m_]:=First[Union[Table[Map[Sort,m/.Rule@@@Table[{Union[Flatten[m]][[i]],Union[Flatten[m]][[perm[[i]]]]},{i,Length[perm]}],{0,2}],{perm,Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]],{n,5}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

Original entry on oeis.org

0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

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

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

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A007716, A255903, A255906, A317073, A281116, A317077, A317532, A317533.

Cf. A317748, A317751, A317755, A317757, A317776.

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[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,allnorm[n]}]],{n,6}]

P(n,k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n,k)={my(p=P(n,k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p,i,y) + polcoef(q,i,y)*sum(j=1, n\i, (-1)^j*binomial(k,j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021

Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021

A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

Original entry on oeis.org

0, 1, 6, 30, 130, 629, 2930, 15019, 78224, 438626, 2548481

Offset: 1

Gus Wiseman, Aug 06 2018

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(3) = 6 strongly normal multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{3}}

Cf. A007716, A035310, A255906, A317073, A281116, A317077, A317078, A317532, A317533.

Cf. A317748, A317751, A317752, A317757, A317775.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]],{n,6}]

a(10)-a(11) from Robert Price, May 08 2021

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

A055884 Euler transform of partition triangle A008284.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 4, 5, 1, 4, 8, 7, 7, 1, 6, 12, 16, 12, 11, 1, 6, 17, 25, 28, 19, 15, 1, 8, 22, 43, 49, 48, 30, 22, 1, 8, 30, 58, 87, 88, 77, 45, 30, 1, 10, 36, 87, 134, 167, 151, 122, 67, 42, 1, 10, 45, 113, 207, 270, 296, 247, 185, 97, 56, 1, 12, 54, 155, 295, 448, 510, 507, 394, 278, 139, 77

Offset: 1

Christian G. Bower, Jun 09 2000

Number of multiset partitions of length-k integer partitions of n. - Gus Wiseman, Nov 09 2018

			From _Gus Wiseman_, Nov 09 2018: (Start)
Triangle begins:
   1
   1   2
   1   2   3
   1   4   4   5
   1   4   8   7   7
   1   6  12  16  12  11
   1   6  17  25  28  19  15
   1   8  22  43  49  48  30  22
   1   8  30  58  87  88  77  45  30
   ...
The fifth row {1, 4, 8, 7, 7} counts the following multiset partitions:
  {{5}}   {{1,4}}     {{1,1,3}}       {{1,1,1,2}}         {{1,1,1,1,1}}
          {{2,3}}     {{1,2,2}}      {{1},{1,1,2}}       {{1},{1,1,1,1}}
         {{1},{4}}   {{1},{1,3}}     {{1,1},{1,2}}       {{1,1},{1,1,1}}
         {{2},{3}}   {{1},{2,2}}     {{2},{1,1,1}}      {{1},{1},{1,1,1}}
                     {{2},{1,2}}    {{1},{1},{1,2}}     {{1},{1,1},{1,1}}
                     {{3},{1,1}}    {{1},{2},{1,1}}    {{1},{1},{1},{1,1}}
                    {{1},{1},{3}}  {{1},{1},{1},{2}}  {{1},{1},{1},{1},{1}}
                    {{1},{2},{2}}
(End)

Alois P. Heinz, Rows n = 1..200, flattened
N. J. A. Sloane, Transforms

Row sums give A001970.
Main diagonal gives A000041.
Columns k=1-2 give: A057427, A052928.
T(n+2,n+1) gives A000070.
T(2n,n) gives A360468.
Cf. A055885, A055886, A360742.
Cf. A000219, A007716, A008284, A255906, A317449, A317532, A317533, A320796, A320801, A320808.

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-1, 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=1..n), n=1..12);  # Alois P. Heinz, Feb 17 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]]]];
Table[Length[Join@@mps/@IntegerPartitions[n,{k}]],{n,5},{k,n}] (* Gus Wiseman, Nov 09 2018 *)

A318565 Number of multiset partitions of multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 6, 27, 169, 1029, 7817, 61006, 547537, 5202009, 54506262, 606311524, 7299051826, 92985064466, 1264720212352, 18137495642192, 275078184766323, 4379514178076452, 73235806332442156, 1280229713195027792, 23381809052104639236, 444740694108284116235, 8801030741502964613534

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities.

			The a(2) = 6 multiset partitions of multiset partitions:
  {{{1,1}}}
  {{{1,2}}}
  {{{1},{1}}}
  {{{1},{2}}}
  {{{1}},{{1}}}
  {{{1}},{{2}}}

Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318564, A318566.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Sum[Length[mps[m]],{m,Join@@mps/@strnorm[n]}],{n,6}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); StronglyNormalLabelingsSeq(sExp(sExp(A))-1)} \\ Andrew Howroyd, Dec 30 2020

Terms a(9) and beyond from Andrew Howroyd, Dec 30 2020

A326211 Number of unsortable normal multiset partitions of weight n.

Original entry on oeis.org

0, 0, 0, 1, 17, 170, 1455, 11678, 92871, 752473

Offset: 0

Gus Wiseman, Jun 19 2019

A multiset partition is normal if it covers an initial interval of positive integers. It is unsortable if no permutation has an ordered concatenation, or equivalently if the concatenation of its lexicographically-ordered parts is not weakly increasing. For example, the multiset partition {{1,2},{1,1,1},{2,2,2}} is sortable because the permutation ((1,1,1),(1,2),(2,2,2)) has concatenation (1,1,1,1,2,2,2,2), which is weakly increasing.

			The a(3) = 1 and a(4) = 17 multiset partitions:
  {{1,3},{2}}  {{1,1,3},{2}}
               {{1,2},{1,2}}
               {{1,2},{1,3}}
               {{1,2,3},{2}}
               {{1,2,4},{3}}
               {{1,3},{2,2}}
               {{1,3},{2,3}}
               {{1,3},{2,4}}
               {{1,3,3},{2}}
               {{1,3,4},{2}}
               {{1,4},{2,3}}
               {{1},{1,3},{2}}
               {{1},{2,4},{3}}
               {{1,3},{2},{2}}
               {{1,3},{2},{3}}
               {{1,3},{2},{4}}
               {{1,4},{2},{3}}

Unsortable set partitions are A058681.
Sortable normal multiset partitions are A326212.
Non-crossing normal multiset partitions are A324171.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000108, A016098, A255906, A324170.
Cf. A326209, A326210, A326243, A326250, A326255, A326256.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
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]]]];
Table[Length[Select[Sort[#,lexsort]&/@Join@@mps/@allnorm[n],!OrderedQ[Join@@#]&]],{n,0,5}]

A255906(n) = a(n) + A326212(n).

cp's OEIS Frontend

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

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A317532 Regular triangle read by rows: T(n,k) is the number of multiset partitions of normal multisets of size n into k blocks, where a multiset is normal if it spans an initial interval of positive integers.

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A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

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A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

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Mathematica

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A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

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Mathematica

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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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Maple

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