OFFSET
1,4
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
1 2 3
1 2
2 3
EXAMPLE
The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
3 3 2 3 2 1 1 3 2 1
4 3 3 2 2 3 2 1 1 1
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]], UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]}, Length[spsu[ptnverts[y], ptnpos[y]]]], {n, 30}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 19 2018
STATUS
approved