A324326 - OEIS

%I #5 Feb 22 2019 21:16:46

%S 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,

%T 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,

%U 0,0,0,0,0,0,103,0,0,0,0,0,0,0,75

%N Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

%C 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}.

%C 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.

%F a(n) + A324325(n) = A318284(n).

%e The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:

%e   {{1,3},{1,2,2,4}}

%e   {{2,4},{1,1,2,3}}

%e   {{1,1,3},{2,2,4}}

%e   {{1,2,3},{1,2,4}}

%e   {{1},{1,3},{2,2,4}}

%e   {{1},{2,4},{1,2,3}}

%e   {{2},{1,3},{1,2,4}}

%e   {{2},{1,1,3},{2,4}}

%e   {{1,2},{1,3},{2,4}}

%e   {{1},{2},{1,3},{2,4}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];

%t Table[Length[Select[facs[n],croXQ[primeMS/@#]&]],{n,Array[Times@@Prime/@nrmptn[#]&,30]}]

%Y Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

%Y Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324325.

%K nonn

%O 1,24

%A _Gus Wiseman_, Feb 22 2019