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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.

%I A321731 #5 Nov 19 2018 07:21:55
%S A321731 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 A321731 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 A321731 0,0,0,0,0,0,0,86,0,0,36,0,0,0,0,22,60,0,0
%N 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.
%C A321731 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321731 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 A321731   1 2 3
%C A321731   1 2
%C A321731   2 3
%e A321731 The a(30) = 12 partitions of the Young diagram of (321) into vertical sections of sizes (321), shown as colorings by positive integers:
%e A321731   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
%e A321731   1 2     1 3     2 1     3 1     1 2     1 3
%e A321731   1       1       1       1       2       3
%e A321731 .
%e A321731   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3   1 2 3
%e A321731   2 1     3 1     2 3     3 2     2 3     3 2
%e A321731   2       3       2       2       3       3
%t A321731 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321731 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321731 ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t A321731 ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
%t A321731 Table[With[{y=Reverse[primeMS[n]]},Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]]],{n,30}]
%Y A321731 Cf. A000110, A000258, A000700, A000701, A056239, A122111, A321649, A321728, A321729, A321730, A321737, A321738.
%K A321731 nonn
%O A321731 1,6
%A A321731 _Gus Wiseman_, Nov 18 2018