A321738 - OEIS

%I #6 Nov 20 2018 12:21:04

%S 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,

%T 60,1,52,31,8,73,114,1,9,43,77,1,115,1,37,235,10,1,151,209,175,57,50,

%U 1,409,136,141,73,11,1,295,1,12,543,203,229,198,1,65,91

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

%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 partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:

%C   1 2 3

%C   1 2

%C   2 3

%e The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:

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

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

%e   4     3     3     2     2     3     2     1     1     1

%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[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];

%t Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]

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

%K nonn,more

%O 1,4

%A _Gus Wiseman_, Nov 19 2018