A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91

Offset: 1

Author

Gus Wiseman, Nov 19 2018

Keywords

• nonn
• more

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

Examples

			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
		

Crossrefs

Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.

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[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]