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%I #5 Nov 19 2018 07:21:48
%S 1,1,1,3,8,23,79,303,1294,5934,29385,156232,884893
%N Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.
%C 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 1 2 3
%C 1 2
%C 2 3
%e The a(5) = 23 partitions of Young diagrams of integer partitions of 5 into vertical sections of the same sizes as the parts of the original partition, shown as colorings by positive integers:
%e 1 2 3 1 2 3 1 2 3
%e 1 2 3
%e 1 2 3
%e .
%e 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%e 1 2 1 3 1 3 2 1 3 1 3 1 2 3 3 2 2 3 3 2
%e 3 2 3 3 2 3 1 1 3 3
%e .
%e 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%e 1 3 3 2 3 3 3 3 3
%e 3 1 4 3 2 4 3 4 4
%e 4 4 1 4 4 2 4 3 4
%e .
%e 1
%e 2
%e 3
%e 4
%e 5
%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[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
%t Table[Sum[Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]],{y,IntegerPartitions[n]}],{n,5}]
%Y Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Nov 18 2018
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