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%I #9 Feb 04 2019 07:37:34
%S 1,1,0,1,1,1,0,0,1,0,2,1,0,0,0,0,1,1,3,1,0,2,0,4,1,0,0,0,3,1,0,0,0,0,
%T 0,0,1,0,2,2,5,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,4,1,0,0,0,6,0,6,1,1,
%U 3,4,6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.
%C Row n has length A000041(A056239(n)).
%C A vertical section is a partial Young diagram with at most one square in each row.
%e Triangle begins:
%e 1
%e 1
%e 0 1
%e 1 1
%e 0 0 1
%e 0 2 1
%e 0 0 0 0 1
%e 1 3 1
%e 0 2 0 4 1
%e 0 0 0 3 1
%e 0 0 0 0 0 0 1
%e 0 2 2 5 1
%e 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 4 1
%e 0 0 0 6 0 6 1
%e 1 3 4 6 1
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 4 10 4 8 1
%e The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
%e T(12,7) = 0:
%e .
%e T(12,9) = 2: 1 2 1 2
%e 1 2
%e 2 1
%e .
%e T(12,10) = 2: 1 2 1 2
%e 2 1
%e 2 1
%e .
%e T(12,12) = 5: 1 2 1 2 1 2 1 2 1 2
%e 3 2 3 1 3
%e 3 3 2 3 1
%e .
%e T(12,16) = 1: 1 2
%e 3
%e 4
%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]]},Table[Length[Select[spsu[ptnverts[y],ptnpos[y]],Sort[Length/@#]==primeMS[k]&]],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]],{n,18}]
%Y Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.
%K nonn,tabf
%O 1,11
%A _Gus Wiseman_, Nov 19 2018
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