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

%I A321738 #6 Nov 20 2018 12:21:04
%S A321738 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 A321738 60,1,52,31,8,73,114,1,9,43,77,1,115,1,37,235,10,1,151,209,175,57,50,
%U A321738 1,409,136,141,73,11,1,295,1,12,543,203,229,198,1,65,91
%N A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.
%C A321738 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A321738 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 A321738   1 2 3
%C A321738   1 2
%C A321738   2 3
%e A321738 The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
%e A321738   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
%e A321738   3     3     2     3     2     1     1     3     2     1
%e A321738   4     3     3     2     2     3     2     1     1     1
%t A321738 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321738 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321738 ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t A321738 ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
%t A321738 Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]
%Y A321738 Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.
%K A321738 nonn,more
%O A321738 1,4
%A A321738 _Gus Wiseman_, Nov 19 2018