A330728 -id:A330728 - cp's OEIS frontend

A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

1, 1, 2, 6, 43, 440, 7158, 151414

Offset: 0

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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

			The a(2) = 2 and a(3) = 6 multisystems:
  {1,1}  {{1},{1,1}}
  {1,2}  {{1},{1,2}}
         {{1},{2,3}}
         {{2},{1,1}}
         {{2},{1,3}}
         {{3},{1,2}}
The a(4) = 43 multisystems (commas and outer brackets elided):
  {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}}
  {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}}
                 {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}}
                 {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}}
                 {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}}
                                {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}}
                                {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}}
                                               {{2}}{{3}{11}} {{2}}{{3}{14}}
                                               {{23}}{{1}{1}} {{23}}{{1}{4}}
                                               {{3}}{{1}{12}} {{2}}{{4}{13}}
                                               {{3}}{{2}{11}} {{24}}{{1}{3}}
                                                              {{3}}{{1}{24}}
                                                              {{3}}{{2}{14}}
                                                              {{3}}{{4}{12}}
                                                              {{34}}{{1}{2}}
                                                              {{4}}{{1}{23}}
                                                              {{4}}{{2}{13}}
                                                              {{4}}{{3}{12}}

The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330475.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The (weakly) normal version is A330676.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of strongly normal multisets are A035310.
Series-reduced rooted trees with strongly normal leaves are A316652.
Cf. A000311, A000669, A001055, A001678, A005121, A005804, A316651, A318812, A330467, A330474, A330625, A330628, A330664, A330677, A330679.

strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 8, 70, 1012, 21944, 665708, 26917492, 1399033348, 90878863352, 7214384973908, 687197223963640, 77354805301801012, 10158257981179981304, 1539156284259756811748, 266517060496258245459352, 52301515332984084095078308, 11546416513975694879642736152

Offset: 0

Gus Wiseman, Dec 30 2019

nonn

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

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

			The a(0) = 1 through a(3) = 8 multisystems:
  {}  {1}  {1,1}  {{1},{1,1}}
           {1,2}  {{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

Row sums of A330778.
The case with all atoms equal is A000111.
The case with all atoms different is A006472.
The version allowing all depths is A330655.
The unlabeled version is A330663.
The version where the atoms are the prime indices of n is A330665.
The strongly normal version is A330675.
The version where the degrees are the prime indices of n is A330728.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000669, A001055, A005121, A005804, A318812, A330469, A330474, A330654, A330664, A330677, A330679.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
seq(n)={concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k))))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2019

A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 4, 5, 5, 7, 16, 16, 27, 2, 61, 33, 272, 27, 123, 61, 1385, 27, 78, 272, 95, 123, 7936, 362

Offset: 1

Gus Wiseman, Dec 28 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Non-isomorphic representatives of the a(n) multisystems for n = 2, 3, 6, 9, 10, 12 (commas and outer brackets elided):
  1  11  {1}{12}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
         {2}{11}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
                  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                  {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                                  {{2}}{{1}{11}}  {{2}}{{1}{13}}
                                                  {{2}}{{3}{11}}
                                                  {{23}}{{1}{1}}

The non-maximal version is A330666.
The case of constant or strict atoms is A000111.
Labeled versions are A330728, A330665 (prime indices), and A330675 (strongly normal).
Non-isomorphic multiset partitions whose degrees are the prime indices of n are A318285.
Cf. A004114, A005121, A007716, A048816, A141268, A306186, A318846, A318848, A330470, A330474, A330663.

For n > 1, a(2^n) = a(prime(n)) = A000111(n - 1).

A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 2, 10, 11, 20, 15, 90, 51, 80, 6, 468, 93, 2910, 80, 521, 277, 20644, 80, 334, 1761, 393, 521, 165874, 1374

Offset: 1

Gus Wiseman, Dec 30 2019

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 10 multisystems (commas and outer brackets elided):
    1  11  12  111      112      1111            123      1122
               {1}{11}  {1}{12}  {1}{111}        {1}{23}  {1}{122}
                        {2}{11}  {11}{11}                 {11}{22}
                                 {1}{1}{11}               {12}{12}
                                 {{1}}{{1}{11}}           {1}{1}{22}
                                 {{11}}{{1}{1}}           {1}{2}{12}
                                                          {{1}}{{1}{22}}
                                                          {{11}}{{2}{2}}
                                                          {{1}}{{2}{12}}
                                                          {{12}}{{1}{2}}
Non-isomorphic representatives of the a(12) = 15 multisystems:
  {1,1,2,3}
  {{1},{1,2,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{2},{1,1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{{1}},{{1},{2,3}}}
  {{{1,1}},{{2},{3}}}
  {{{1}},{{2},{1,3}}}
  {{{1,2}},{{1},{3}}}
  {{{2}},{{1},{1,3}}}
  {{{2}},{{3},{1,1}}}
  {{{2,3}},{{1},{1}}}

The labeled version is A318846.
The maximum-depth version is A330664.
Unlabeled balanced reduced multisystems by weight are A330474.
The case of constant or strict atoms is A318813.
Cf. A000669, A005121, A007716, A048816, A141268, A306186, A317791, A318812, A318849, A330470, A330475, A330655, A330728.

a(2^n) = a(prime(n)) = A318813(n).

A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 1, 7, 7, 1, 5, 5, 1, 5, 9, 5, 1, 9, 11, 1, 9, 28, 36, 16, 1, 10, 24, 16, 1, 14, 38, 27, 1, 13, 18, 1, 13, 69, 160, 164, 61, 1, 24, 79, 62, 1, 20, 160, 580, 1022, 855, 272, 1, 19, 59, 45, 1, 27, 138, 232, 123, 1, 17, 77, 121, 61

Offset: 2

Gus Wiseman, Jan 04 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Triangle begins:
   {}
   1
   1
   1   1
   1   2
   1   3   2
   1   3
   1   7   7
   1   5   5
   1   5   9   5
   1   9  11
   1   9  28  36  16
   1  10  24  16
   1  14  38  27
   1  13  18
   1  13  69 160 164  61
   1  24  79  62
For example, row n = 12 counts the following multisystems:
  {1,1,2,3}  {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,3}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{3},{1,1,2}}    {{{1}},{{3},{1,2}}}
             {{1},{1},{2,3}}  {{{1,3}},{{1},{2}}}
             {{1},{2},{1,3}}  {{{2}},{{1},{1,3}}}
             {{1},{3},{1,2}}  {{{2}},{{3},{1,1}}}
             {{2},{3},{1,1}}  {{{2,3}},{{1},{1}}}
                              {{{3}},{{1},{1,2}}}
                              {{{3}},{{2},{1,1}}}

Row sums are A318846.
Final terms in each row are A330728.
Row prime(n) is row n of A330784.
Row 2^n is row n of A008826.
Row n is row A181821(n) of A330667.
Column k = 3 is A318284(n) - 2 for n > 2.
Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(2^n,k) = A008826(n,k).

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187

Offset: 0

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(1) = 1 through a(5) = 17 multisystems (commas elided):
  {1}  {2}   {3}        {4}               {5}
       {11}  {12}       {22}              {23}
             {{1}{11}}  {13}              {14}
                        {{1}{12}}         {{1}{13}}
                        {{2}{11}}         {{1}{22}}
                        {{{1}}{{1}{11}}}  {{2}{12}}
                        {{{11}}{{1}{1}}}  {{3}{11}}
                                          {{{1}}{{1}{12}}}
                                          {{{11}}{{1}{2}}}
                                          {{{1}}{{2}{11}}}
                                          {{{12}}{{1}{1}}}
                                          {{{2}}{{1}{11}}}
                                          {{{{1}}}{{{1}}{{1}{11}}}}
                                          {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}

The case with all atoms equal to 1 is A000111.
The non-maximal version is A330679.
A tree version is A320160.
Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

cp's OEIS Frontend

A330675 Number of balanced reduced multisystems of maximum depth whose atoms constitute a strongly normal multiset of size n.

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A330676 Number of balanced reduced multisystems of weight n and maximum depth whose atoms cover an initial interval of positive integers.

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A330664 Number of non-isomorphic balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

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A330666 Number of non-isomorphic balanced reduced multisystems whose degrees (atom multiplicities) are the weakly decreasing prime indices of n.

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A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

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A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

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