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%I #11 Jul 23 2018 09:15:18
%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,7,2,2,
%T 2,8,1,2,2,7,1,5,1,4,4,2,1,10,2,4,2,4,1,7,2,7,2,2,1,9,1,2,4,11,2,5,1,
%U 4,2,5,1,12,1,2,4,4,2,5,1,11,5,2,1,10,2
%N In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H Alois P. Heinz, <a href="/A317141/b317141.txt">Table of n, a(n) for n = 1..65536</a>
%e The a(24) = 6 partitions coarser than or equal to (2111) are (2111), (311), (221), (32), (41), (5), with Heinz numbers 24, 20, 18, 15, 14, 11.
%p g:= l-> `if`(l=[], {[]}, (t-> map(sort, map(x->
%p [seq(subsop(i=x[i]+t, x), i=1..nops(x)),
%p [x[], t]][], g(subsop(-1=[][], l)))))(l[-1])):
%p a:= n-> nops(g(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 22 2018
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t ptncaps[ptn_]:=Union[Sort/@Apply[Plus,mps[ptn],{2}]];
%t Table[Length[ptncaps[primeMS[n]]],{n,100}]
%Y Cf. A002846, A056239, A213427, A215366, A265947, A296150, A296150, A299201, A300383, A317142, A317143.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jul 22 2018
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