OFFSET
1,3
COMMENTS
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 multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
EXAMPLE
The sequence of antichains begins:
2: {{1}}
3: {{1,1}}
3: {{1},{1}}
4: {{1,2}}
5: {{1,1,1}}
5: {{1},{1},{1}}
6: {{1,1,2}}
7: {{1,1,1,1}}
7: {{1,1},{1,1}}
7: {{1},{1},{1},{1}}
8: {{1,2,3}}
9: {{1,1,2,2}}
10: {{1,1,1,2}}
10: {{1,1},{1,2}}
11: {{1,1,1,1,1}}
11: {{1},{1},{1},{1},{1}}
12: {{1,1,2,3}}
12: {{1,2},{1,3}}
13: {{1,1,1,1,1,1}}
13: {{1,1,1},{1,1,1}}
13: {{1,1},{1,1},{1,1}}
13: {{1},{1},{1},{1},{1},{1}}
14: {{1,1,1,1,2}}
14: {{1,2},{1,1,1}}
15: {{1,1,1,2,2}}
15: {{1,1},{1,2,2}}
16: {{1,2,3,4}}
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[Times@@Prime/@nrmptn[n]], And[zensity[#]==-1, Length[zsm[#]]==1, Select[Tuples[#, 2], UnsameQ@@#&&Divisible@@#&]=={}]&]], {n, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 01 2018
STATUS
approved