cp's OEIS Frontend

A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

%I A324326 #5 Feb 22 2019 21:16:46
%S A324326 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 A324326 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 A324326 0,0,0,0,0,0,103,0,0,0,0,0,0,0,75
%N A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.
%C A324326 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 A324326 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 A324326 a(n) + A324325(n) = A318284(n).
%e A324326 The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:
%e A324326   {{1,3},{1,2,2,4}}
%e A324326   {{2,4},{1,1,2,3}}
%e A324326   {{1,1,3},{2,2,4}}
%e A324326   {{1,2,3},{1,2,4}}
%e A324326   {{1},{1,3},{2,2,4}}
%e A324326   {{1},{2,4},{1,2,3}}
%e A324326   {{2},{1,3},{1,2,4}}
%e A324326   {{2},{1,1,3},{2,4}}
%e A324326   {{1,2},{1,3},{2,4}}
%e A324326   {{1},{2},{1,3},{2,4}}
%t A324326 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A324326 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A324326 croXQ[stn_]:=MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324326 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
%t A324326 Table[Length[Select[facs[n],croXQ[primeMS/@#]&]],{n,Array[Times@@Prime/@nrmptn[#]&,30]}]
%Y A324326 Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.
%Y A324326 Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324325.
%K A324326 nonn
%O A324326 1,24
%A A324326 _Gus Wiseman_, Feb 22 2019