A330676 -id:A330676 - cp's OEIS frontend

A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

1, 1, 2, 12, 138, 2652, 78106, 3256404, 182463296, 13219108288, 1202200963522, 134070195402644, 17989233145940910, 2858771262108762492, 530972857546678902490, 113965195745030648131036, 27991663753030583516229824, 7800669355870672032684666900, 2448021231611414334414904013956

Offset: 0

Gus Wiseman, Dec 27 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.

			The a(0) = 1 through a(3) = 12 multisystems:
  {}  {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

Row sums of A330776.
The unlabeled version is A330474.
The strongly normal case is A330475.
The tree version is A330654.
The maximum-depth case is A330676.
The case where the atoms are all different is A005121.
The case where the atoms are all equal is A318813.
Multiset partitions of normal multisets are A255906.
Series-reduced rooted trees with normal leaves are A316651.
Cf. A000311, A000669, A001678, A213427, A318812, A330675, 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, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-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 2019

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2019

A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

Original entry on oeis.org

1, 1, 2, 4, 20, 140, 1411

Offset: 0

Gus Wiseman, Dec 27 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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
  {1,1}  {{1},{1,1}}  {{{1}},{{1},{1,1}}}
  {1,2}  {{1},{1,2}}  {{{1,1}},{{1},{1}}}
         {{1},{2,3}}  {{{1}},{{1},{1,2}}}
         {{2},{1,1}}  {{{1,1}},{{1},{2}}}
                      {{{1}},{{1},{2,2}}}
                      {{{1,1}},{{2},{2}}}
                      {{{1}},{{1},{2,3}}}
                      {{{1,1}},{{2},{3}}}
                      {{{1}},{{2},{1,1}}}
                      {{{1,2}},{{1},{1}}}
                      {{{1}},{{2},{1,2}}}
                      {{{1,2}},{{1},{2}}}
                      {{{1}},{{2},{1,3}}}
                      {{{1,2}},{{1},{3}}}
                      {{{1}},{{2},{3,4}}}
                      {{{1,2}},{{3},{4}}}
                      {{{2}},{{1},{1,1}}}
                      {{{2}},{{1},{1,3}}}
                      {{{2}},{{3},{1,1}}}
                      {{{2,3}},{{1},{1}}}

The non-maximal version is A330474.
Labeled versions are A330675 (strongly normal) and A330676 (normal).
The case where the leaves are sets (as opposed to multisets) is A330677.
The case with all atoms distinct is A000111.
The case with all atoms equal is (also) A000111.
Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.

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

Original entry on oeis.org

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

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 16, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 16, 1, 3, 1, 2, 1, 3, 1, 27, 1, 1, 2, 2, 1, 3, 1, 16, 2, 1, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 27 2019

nonn

First differs from A317145 at a(32) = 5, A317145(32) = 4.
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.
Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

			The a(n) multisystems for n = 2, 6, 12, 24, 48:
  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}
              {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}
                           {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}
                           {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}
                           {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}
                                                {{{{1}}},{{{1}},{{2},{1,1}}}}
                                                {{{{1}}},{{{1,2}},{{1},{1}}}}
                                                {{{{1},{1}}},{{{2}},{{1,1}}}}
                                                {{{{1},{1,2}}},{{{1}},{{1}}}}
                                                {{{{1,1}}},{{{2}},{{1},{1}}}}
                                                {{{{1}}},{{{2}},{{1},{1,1}}}}
                                                {{{{1},{2}}},{{{1}},{{1,1}}}}
                                                {{{{1,2}}},{{{1}},{{1},{1}}}}
                                                {{{{2}}},{{{1}},{{1},{1,1}}}}
                                                {{{{2}}},{{{1,1}},{{1},{1}}}}
                                                {{{{2},{1,1}}},{{{1}},{{1}}}}

The last nonzero term in row n of A330667 is a(n).
The chain version is A317145.
The non-maximal version is A318812.
Unlabeled versions are A330664 and A330663.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

primeMS[n_]:=If[n==1,{},Flatten[Cases[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

a(2^n) = A000111(n - 1).

a(product of n distinct primes) = A006472(n).

A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 7, 5, 5, 11, 16, 16, 27, 18, 61, 62, 272, 45, 123, 61, 1385, 105, 152, 272, 501, 211, 7936, 362

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}.

			The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):
  11  {1}{12}  {1}{23}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
      {2}{11}  {2}{13}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
               {3}{12}  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                        {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                        {{2}}{{1}{12}}  {{2}}{{1}{11}}  {{1}}{{3}{12}}
                        {{2}}{{2}{11}}                  {{13}}{{1}{2}}
                        {{22}}{{1}{1}}                  {{2}}{{1}{13}}
                                                        {{2}}{{3}{11}}
                                                        {{23}}{{1}{1}}
                                                        {{3}}{{1}{12}}
                                                        {{3}}{{2}{11}}

The version with distinct atoms is A006472.
The non-maximal version is A318846.
A tree version is A318848, with orderless version A318849.
The unlabeled version is A330664.
Final terms in each row of A330727.
See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).
Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.

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

a(2^n) = A006472(n).

a(prime(n)) = A000111(n - 1).

A330654 Number of series/singleton-reduced rooted trees on normal multisets of size n.

Original entry on oeis.org

1, 1, 2, 12, 112, 1444, 24099, 492434, 11913985

Offset: 0

Gus Wiseman, Dec 26 2019

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).
A finite multiset is normal if it covers an initial interval of positive integers.
First differs from A316651 at a(6) = 24099, A316651(6) = 24086. For example, ((1(12))(2(11))) and ((2(11))(1(12))) are considered identical for A316651 (series-reduced rooted trees), but {{{1},{1,2}},{{2},{1,1}}} and {{{2},{1,1}},{{1},{1,2}}} are different series/singleton-reduced rooted trees.

			The a(0) = 1 through a(3) = 12 trees:
  {}  {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}}

The orderless version is A316651.
The strongly normal case is A330471.
The unlabeled version is A330470.
The balanced version is A330655.
The case with all atoms distinct is A000311.
The case with all atoms equal is A196545.
Normal multiset partitions are A255906.
Cf. A000669, A004114, A005804, A281118, A316651, A330469, A330626, A330676.

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]];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[ssrtrees[s]],{s,allnorm[n]}],{n,0,5}]

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

A330778 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with maximum depth and atoms colored using exactly k colors.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 2, 17, 33, 18, 5, 86, 321, 420, 180, 16, 520, 3306, 7752, 7650, 2700, 61, 3682, 37533, 140172, 238560, 189000, 56700, 272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600, 1385, 278414, 6630909, 53244180, 199775820, 398328480, 435954960, 247665600, 57153600

Offset: 1

Andrew Howroyd, Dec 30 2019

			Triangle begins:
    1;
    1,     1;
    1,     4,      3;
    2,    17,     33,      18;
    5,    86,    321,     420,     180;
   16,   520,   3306,    7752,    7650,    2700;
   61,  3682,  37533,  140172,  238560,  189000,   56700;
  272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600;
  ...

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

Column 1 is A000111.
Main diagonal is A006472.
Row sums are A330676.
Cf. A330776.

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}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}

cp's OEIS Frontend

A330655 Number of balanced reduced multisystems of weight n whose atoms cover an initial interval of positive integers.

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A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.

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

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A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

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A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

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A330654 Number of series/singleton-reduced rooted trees on normal multisets of size 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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A330778 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with maximum depth and atoms colored using exactly k colors.

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