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A321731
Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections of the same sizes as the parts of the original partition.
4
1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 0, 5, 0, 0, 0, 1, 0, 10, 0, 3, 0, 0, 0, 9, 0, 0, 8, 0, 0, 12, 0, 1, 0, 0, 0, 34, 0, 0, 0, 10, 0, 0, 0, 0, 24, 0, 0, 14, 0, 0, 0, 0, 0, 68, 0, 4, 0, 0, 0, 78, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 86, 0, 0, 36, 0, 0, 0, 0, 22, 60, 0, 0
OFFSET
1,6
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 suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
1 2 3
1 2
2 3
EXAMPLE
The a(30) = 12 partitions of the Young diagram of (321) into vertical sections of sizes (321), shown as colorings by positive integers:
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
1 2 1 3 2 1 3 1 1 2 1 3
1 1 1 1 2 3
.
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
2 1 3 1 2 3 3 2 2 3 3 2
2 3 2 2 3 3
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[Join@@Table[Subsets[ptnpos[y], {k}], {k, Reverse[Union[y]]}], UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]}, Length[Select[spsu[ptnverts[y], ptnpos[y]], Function[p, Sort[Length/@p]==Sort[y]]]]], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 18 2018
STATUS
approved