cp's OEIS Frontend

A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

%I A324325 #5 Feb 22 2019 21:16:38
%S A324325 1,1,2,2,3,4,5,5,9,7,7,11,11,12,16,14,15,26,22,21,29,19,30,33,31,30,
%T A324325 66,38,42,52,56,42,47,45,57,82,77,67,77,67,101,98,135,64,137,97,176,
%U A324325 104,109,109,118,105,231,213,97,127,181,139,297,173,385,195,269
%N A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.
%C A324325 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 A324325 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.
%F A324325 a(n) + A324326(n) = A318284(n).
%e A324325 The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
%e A324325   {{1,2,3,4}}
%e A324325   {{1},{2,3,4}}
%e A324325   {{2},{1,3,4}}
%e A324325   {{3},{1,2,4}}
%e A324325   {{4},{1,2,3}}
%e A324325   {{1,2},{3,4}}
%e A324325   {{1,4},{2,3}}
%e A324325   {{1},{2},{3,4}}
%e A324325   {{1},{3},{2,4}}
%e A324325   {{1},{4},{2,3}}
%e A324325   {{2},{3},{1,4}}
%e A324325   {{2},{4},{1,3}}
%e A324325   {{3},{1,2},{4}}
%e A324325   {{1},{2},{3},{4}}
%e A324325 Missing from this list is {{1,3},{2,4}}.
%t A324325 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A324325 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A324325 nonXQ[stn_]:=!MatchQ[stn,{___,{___,x_,___,y_,___},___,{___,z_,___,t_,___},___}/;x<z<y<t||z<x<t<y];
%t A324325 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
%t A324325 Table[Length[Select[facs[n],nonXQ[primeMS/@#]&]],{n,Array[Times@@Prime/@nrmptn[#]&,30]}]
%Y A324325 Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.
%Y A324325 Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.
%K A324325 nonn
%O A324325 1,3
%A A324325 _Gus Wiseman_, Feb 22 2019