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

Author

Gus Wiseman, Nov 18 2018

Keywords

• nonn

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

Examples

			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
		

Crossrefs

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

Programs

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}]