cp's OEIS Frontend

A321730 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.

%I A321730 #5 Nov 19 2018 07:21:48
%S A321730 1,1,1,3,8,23,79,303,1294,5934,29385,156232,884893
%N A321730 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 A321730 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 A321730   1 2 3
%C A321730   1 2
%C A321730   2 3
%e A321730 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 A321730   1 2 3   1 2 3   1 2 3
%e A321730   1       2       3
%e A321730   1       2       3
%e A321730 .
%e A321730   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
%e A321730   1 2   1 3   1 3   2 1   3 1   3 1   2 3   3 2   2 3   3 2
%e A321730   3     2     3     3     2     3     1     1     3     3
%e A321730 .
%e A321730   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
%e A321730   1     3     3     2     3     3     3     3     3
%e A321730   3     1     4     3     2     4     3     4     4
%e A321730   4     4     1     4     4     2     4     3     4
%e A321730 .
%e A321730   1
%e A321730   2
%e A321730   3
%e A321730   4
%e A321730   5
%t A321730 spsu[_,{}]:={{}};spsu[foo_,set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,___}];
%t A321730 ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
%t A321730 ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
%t A321730 Table[Sum[Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]],{y,IntegerPartitions[n]}],{n,5}]
%Y A321730 Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.
%K A321730 nonn,more
%O A321730 0,4
%A A321730 _Gus Wiseman_, Nov 18 2018