A318369 -id:A318369 - cp's OEIS frontend

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 1, 2, 1, 5, 3, 2, 1, 6, 1, 2, 3, 15, 1, 9, 1, 6, 3, 2, 1, 21, 4, 2, 16, 6, 1, 10, 1, 52, 3, 2, 4, 35, 1, 2, 3, 22, 1, 10, 1, 6, 19, 2, 1, 83, 5, 13, 3, 6, 1, 66, 4, 22, 3, 2, 1, 41, 1, 2, 20, 203, 4, 10, 1, 6, 3, 14, 1, 153, 1, 2, 26, 6, 5, 10, 1

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(12) = 6 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A001055, A007716, A049311, A116540, A181821, A188392, A255906.

Cf. A318283, A318284, A318286, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,80}]

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i,2], j, primepi(f[i,1]))))}
count(sig)={my(n=vecsum(sig), s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + O(x*x^n))); s+=prod(i=1, #sig, polcoef(q,sig[i]))*permcount(p)); s/n!}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s<=2, if(#s==1, 1, min(s[1],s[2])+1), count(sig(n))))} \\ Andrew Howroyd, Dec 10 2018

a(n) = A050320(A181821(n)).
From Andrew Howroyd, Dec 10 2018:(Start)
a(p) = 1 for prime(p).
a(prime(i)*prime(j)) = min(i,j) + 1.
a(prime(n)^k) = A188392(n,k). (End)

A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 5, 1, 2, 3, 5, 1, 7, 1, 5, 3, 2, 1, 9, 4, 2, 8, 5, 1, 10

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(12) = 5 set multipartitions of {1,1,2,3}:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{1},{2},{3}}

Cf. A007716, A049311, A116540, A181821, A317791.

Cf. A318283, A318285, A318287, A318360, A318361, A318369, A318371.

a(n) = A318369(A181821(n)).

A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 0, 3, 0, 0, 0, 5, 0, 4, 0, 1, 0, 0, 0, 6, 0, 0, 4, 0, 0, 2

Offset: 1

Gus Wiseman, Aug 24 2018

			Non-isomorphic representatives of the a(24) = 6 strict set multipartitions of {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1},{2},{1,3,4}}
  {{1},{1,2},{3,4}}
  {{2},{1,3},{1,4}}
  {{1},{2},{3},{1,4}}

Cf. A007716, A045778, A049311, A050326, A116539, A116540, A283877, A292432 A292444.

Cf. A318283, A318285, A318287, A318360, A318361, A318362, A318369, A318370.

a(n) = A318370(A181821(n)).

A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 3, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 0, 0, 2, 1, 3, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			Non-isomorphic representatives of the a(180) = 4 strict set multipartitions of {1,1,2,2,3}:
  {{1,2},{1,2,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2},{3},{1,2}}

Cf. A007716, A045778, A049311, A050326, A056239, A112798, A116539, A116540, A292432, A292444, A293511, A318284, A318360, A318369.

A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

Original entry on oeis.org

1, 2, 6, 20, 74, 311, 1401, 6913, 36376, 205421, 1228288, 7786802, 51937607, 364250763, 2673314121, 20504809133, 163844631872, 1361874185139, 11748149246269, 105029750531640, 971403871953460, 9282643841237360, 91519776792040324, 929892817423282068, 9725646244888190337

Offset: 1

Gus Wiseman, Aug 25 2018

nonn

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

			The a(4) = 20 sets of sets:
  {{1,2,3,4}}
  {{1},{1,2,3}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{3,4}}
  {{1},{3},{1,2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

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

Cf. A007716, A049311, A050326, A116540, A283877, A292432, A292444, A318361, A318369, A318370.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=WeighT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 2, 0, 0, 1, 0, 11, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 13, 1, 1, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 0, 1, 41, 0, 0, 0, 0, 0, 1, 0, 20, 0, 0, 2, 0, 0, 0, 0, 6, 16, 0, 0, 1, 0

Offset: 1

Gus Wiseman, Nov 25 2018

nonn

This multiset (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(90) = 7 set multipartitions of {1,1,1,2,2,3,3,4} with no singletons:
  {{1,2},{1,2},{1,3},{3,4}}
  {{1,2},{1,3},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{1,2,3,4}}
  {{1,2},{1,2,3},{1,3,4}}
  {{1,3},{1,2,3},{1,2,4}}
  {{1,4},{1,2,3},{1,2,3}}

Cf. A001055, A007716, A049311, A056156, A116540, A181821, A255906, A304382.

Cf. A318284, A318286, A318360, A318361, A318362, A318369.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqnopfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqnopfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],!PrimeQ[#]&&SquareFreeQ[#]&]}]];
Table[Length[sqnopfacs[Times@@Prime/@nrmptn[n]]],{n,30}]

cp's OEIS Frontend

A318360 Number of set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

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A318362 Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.

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A318371 Number of non-isomorphic strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

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A318370 Number of non-isomorphic strict set multipartitions (sets of sets) of the multiset of prime indices of n.

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A318402 Number of sets of nonempty sets whose multiset union is a strongly normal multiset of size n.

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A322076 Number of set multipartitions (multisets of sets) with no singletons, of a multiset whose multiplicities are the prime indices of n.

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