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A318361
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Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
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18
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1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0
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OFFSET
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1,4
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LINKS
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FORMULA
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EXAMPLE
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The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
{{1},{1,2,3,4}}
{{1,2},{1,3,4}}
{{1,3},{1,2,4}}
{{1,4},{1,2,3}}
{{1},{2},{1,3,4}}
{{1},{3},{1,2,4}}
{{1},{4},{1,2,3}}
{{1},{1,2},{3,4}}
{{1},{1,3},{2,4}}
{{1},{1,4},{2,3}}
{{2},{1,3},{1,4}}
{{3},{1,2},{1,4}}
{{4},{1,2},{1,3}}
{{1},{2},{3},{1,4}}
{{1},{2},{4},{1,3}}
{{1},{3},{4},{1,2}}
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MATHEMATICA
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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, 90}]
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PROG
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(PARI)
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(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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