A316974 -id:A316974 - cp's OEIS frontend

A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 6, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 18 2018

nonn

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes. For example, in 60 = (2*30) the primes {3, 5} are equivalent but {2, 3} and {2, 5} are not.

			The a(24) = 5 factorizations are (2*3*4), (2*12), (3*8), (4*6), (24).
The a(36) = 4 factorizations are (2*3*6), (2*18), (3*12), (4*9).

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091, A162247, A281116.

Cf. A316974, A316978, A316980, A316981.

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]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[facs[n],And[UnsameQ@@#,UnsameQ@@dual[primeMS/@#]]&]],{n,100}]

a(prime^n) = A000009(n).

A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 2, 5, 28, 277, 3985, 76117, 1833187, 53756682, 1871041538, 75809298105, 3521419837339, 185235838688677, 10923147890901151, 715989783027216302, 51793686238309903860, 4109310551278549543317, 355667047514571431358297, 33422937748872646130124797

Offset: 0

Gus Wiseman, Jul 17 2018

nonn

Note that all connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} are strict except for (123...n)(123...n).

			The a(2) = 5 connected multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (12)(12), (1)(2)(12). The multiset partitions (11)(22), (1)(1)(22), (2)(2)(11), (1)(1)(2)(2) are not connected.

Cf. A001055, A007716, A007717, A007719, A020555, A045778, A061742, A094574, A162247, A316974.

nn=10;
ser=Exp[-3/2+Exp[x]/2]*Sum[Exp[Binomial[n+1,2]*x]/n!,{n,0,3*nn}];
Round/@(CoefficientList[Series[1+Log[ser],{x,0,nn}],x]*Array[Factorial,nn+1,0]) (* based on Jean-François Alcover after Vladeta Jovovic *)
(*second program *)
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[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]]]];
Length/@Table[Select[mps[Ceiling[Range[1/2,n,1/2]]],Length[csm[#]]==1&],{n,4}]

Logarithmic transform of A020555.

A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows and no equal columns.

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 6 strict multiset partitions with no equivalent vertices (first column) and their duals (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(2)(3) <-> (1)(2)(3)

Cf. A000009, A001055, A007716, A007717, A020555, A045778.

Cf. A316974, A316978, A316979, A316980, A316983.

A316892 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices.

Original entry on oeis.org

1, 1, 3, 9, 24, 69, 211, 654

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of unlabeled graphs with n edges, allowing loops, with no equivalent vertices (two vertices are equivalent if in every edge the multiplicity of the first is equal to the multiplicity of the second). For example, non-isomorphic representatives of the a(2) = 3 multigraphs are {(1,2),(1,3)}, {(1,1),(1,2)}, {(1,1),(2,2)}.

			Non-isomorphic representatives of the a(3) = 9 strict multiset partitions:
  (112)(233)
  (1)(2)(1233)
  (1)(12)(233)
  (2)(11)(233)
  (11)(22)(33)
  (12)(13)(23)
  (1)(2)(3)(123)
  (1)(2)(12)(33)
  (1)(2)(13)(23)

Cf. A007716, A007717, A020555, A050535, A053419, A094574, A316974.

a(6)-a(7) from Andrew Howroyd, Feb 07 2020

A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.

Original entry on oeis.org

1, 12, 575, 66080, 13830706, 4566898564, 2181901435364, 1422774451251512, 1213875872220833664, 1312273759143855989808, 1752860078230602866012288, 2834766624822130489716563008, 5458358420687156358967526721408, 12339106957086349462329140342122112

Offset: 1

Gus Wiseman, Jul 17 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(2) = 12 trees are (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122).

Cf. A000081, A000669, A007717, A020555, A050535, A094574, A292504, A316655, A316972, A316974.

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]]]];
gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[gro[Ceiling[Range[1/2,n,1/2]]]],{n,4}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(2*n), vars=vector(2*n-2,i,sv(2+i))); v[1]=sv(1); for(n=2, #v, v[n] = substvec(polcoef( sExp(x*Ser(v[1..n])), n ), vars[1..n-2], vector(n-2))); sCartProd(x*Ser(v), 1/(1-x^2*symGroupCycleIndex(2)) + O(x*x^(2*n)))}
seq(n)={my(p=substvec(cycleIndexSeries(n), [sv(1), sv(2)], [1,1])); vector(n, n, polcoef(p,2*n))} \\ Andrew Howroyd, Jan 02 2021

a(n) = A292505(A061742(n)). - Andrew Howroyd, Nov 19 2018

Terms a(6) and beyond from Andrew Howroyd, Jan 02 2021

cp's OEIS Frontend

A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

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A316972 Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.

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A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

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A316892 Number of non-isomorphic strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} with no equivalent vertices.

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A316977 Number of series-reduced rooted trees whose leaves are {1, 1, 2, 2, 3, 3, ..., n, n}.

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