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}.
A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} where x < z < y < t or z < x < t < y.
EXAMPLE
The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
{{1,2,3,4}}
{{1},{2,3,4}}
{{2},{1,3,4}}
{{3},{1,2,4}}
{{4},{1,2,3}}
{{1,2},{3,4}}
{{1,4},{2,3}}
{{1},{2},{3,4}}
{{1},{3},{2,4}}
{{1},{4},{2,3}}
{{2},{3},{1,4}}
{{2},{4},{1,3}}
{{3},{1,2},{4}}
{{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.
MATHEMATICA
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]]}]];
nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, Reverse[primeMS[n]]];
Table[Length[Select[facs[n], nonXQ[primeMS/@#]&]], {n, Array[Times@@Prime/@nrmptn[#]&, 30]}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 22 2019
STATUS
approved