A321731 - OEIS

%I #5 Nov 19 2018 07:21:55

%S 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,

%T 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,

%U 0,0,0,0,0,0,0,86,0,0,36,0,0,0,0,22,60,0,0

%N 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.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C 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:

%C   1 2 3

%C   1 2

%C   2 3

%e The a(30) = 12 partitions of the Young diagram of (321) into vertical sections of sizes (321), shown as colorings by positive integers:

%e   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3

%e   1 2     1 3     2 1     3 1     1 2     1 3

%e   1       1       1       1       2       3

%e .

%e   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3

%e   2 1     3 1     2 3     3 2     2 3     3 2

%e   2       3       2       2       3       3

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];

%t ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];

%t ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];

%t Table[With[{y=Reverse[primeMS[n]]},Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]]],{n,30}]

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

%K nonn

%O 1,6

%A _Gus Wiseman_, Nov 18 2018