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A321854 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.

%I A321854 #9 Feb 04 2019 07:37:34
%S A321854 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 A321854 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 A321854 3,4,6,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A321854 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 A321854 Row n has length A000041(A056239(n)).
%C A321854 A vertical section is a partial Young diagram with at most one square in each row.
%e A321854 Triangle begins:
%e A321854   1
%e A321854   1
%e A321854   0  1
%e A321854   1  1
%e A321854   0  0  1
%e A321854   0  2  1
%e A321854   0  0  0  0  1
%e A321854   1  3  1
%e A321854   0  2  0  4  1
%e A321854   0  0  0  3  1
%e A321854   0  0  0  0  0  0  1
%e A321854   0  2  2  5  1
%e A321854   0  0  0  0  0  0  0  0  0  0  1
%e A321854   0  0  0  0  0  4  1
%e A321854   0  0  0  6  0  6  1
%e A321854   1  3  4  6  1
%e A321854   0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
%e A321854   0  0  4 10  4  8  1
%e A321854 The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
%e A321854   T(12,7) = 0:
%e A321854 .
%e A321854   T(12,9) = 2:    1 2   1 2
%e A321854                   1     2
%e A321854                   2     1
%e A321854 .
%e A321854   T(12,10) = 2:   1 2   1 2
%e A321854                   2     1
%e A321854                   2     1
%e A321854 .
%e A321854   T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2
%e A321854                   3     2     3     1     3
%e A321854                   3     3     2     3     1
%e A321854 .
%e A321854   T(12,16) = 1:   1 2
%e A321854                   3
%e A321854                   4
%t A321854 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A321854 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321854 ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t A321854 ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
%t A321854 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 A321854 Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.
%K A321854 nonn,tabf
%O A321854 1,11
%A A321854 _Gus Wiseman_, Nov 19 2018