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A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n. 4
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 36, 0, 14, 0, 0, 0, 25, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 75 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,24
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...}} with x < z < y < t or z < x < t < y.
LINKS
FORMULA
a(n) + A324325(n) = A318284(n).
EXAMPLE
The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:
{{1,3},{1,2,2,4}}
{{2,4},{1,1,2,3}}
{{1,1,3},{2,2,4}}
{{1,2,3},{1,2,4}}
{{1},{1,3},{2,2,4}}
{{1},{2,4},{1,2,3}}
{{2},{1,3},{1,2,4}}
{{2},{1,1,3},{2,4}}
{{1,2},{1,3},{2,4}}
{{1},{2},{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]]}]];
croXQ[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], croXQ[primeMS/@#]&]], {n, Array[Times@@Prime/@nrmptn[#]&, 30]}]
CROSSREFS
Sequence in context: A151795 A008437 A366076 * A341743 A348945 A151755
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 22 2019
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)