A317141 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.

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, 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, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 10, 2

Offset: 1

Author

Gus Wiseman, Jul 22 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A002846, A056239, A213427, A215366, A265947, A296150, A296150, A299201, A300383, A317142, A317143.

Programs

Maple

g:= l-> `if`(l=[], {[]}, (t-> map(sort, map(x->
        [seq(subsop(i=x[i]+t, x), i=1..nops(x)),
        [x[], t]][], g(subsop(-1=[][], l)))))(l[-1])):
a:= n-> nops(g(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 22 2018

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
ptncaps[ptn_]:=Union[Sort/@Apply[Plus,mps[ptn],{2}]];
Table[Length[ptncaps[primeMS[n]]],{n,100}]