cp's OEIS Frontend

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

%I A318560 #8 Aug 29 2018 16:49:47
%S A318560 1,1,2,2,3,4,5,3,8,7,7,8,11,12,15,5,15,17,22,14,27,19,30,13,27,30,33,
%T A318560 26,42,37,56,7,44,45,51,34,77,67,72,25
%N A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.
%C A318560 A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.
%C A318560 The prime indices of n are the n-th row of A296150.
%e A318560 The a(18) = 17 combinatory separations of {1,1,2,2,3}:
%e A318560   {11223}
%e A318560   {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
%e A318560   {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
%e A318560   {1,1,1,11} {1,1,1,12}
%e A318560   {1,1,1,1,1}
%t A318560 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A318560 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A318560 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A318560 normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
%t A318560 Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]
%Y A318560 Cf. A007716, A056239, A255906, A269134, A296150, A305936.
%Y A318560 Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.
%K A318560 nonn,more
%O A318560 1,3
%A A318560 _Gus Wiseman_, Aug 28 2018