%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