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.

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

Links

Table of n, a(n) for n=1..83.

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}]

Crossrefs

Cf. A000110, A000258, A000700, A000701, A056239, A122111, A321649, A321728, A321729, A321730, A321737, A321738.

Sequence in context: A115723 A238160 A178524 * A358135 A212357 A114525

Adjacent sequences: A321728 A321729 A321730 * A321732 A321733 A321734

Keyword

nonn

Author

Gus Wiseman, Nov 18 2018

Status

approved