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.

1, 1, 1, 3, 8, 23, 79, 303, 1294, 5934, 29385, 156232, 884893

Offset

0,4

Comments

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:

1 2 3

1 2

2 3

Links

Table of n, a(n) for n=0..12.

Example

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:
  1 2 3   1 2 3   1 2 3
  1       2       3
  1       2       3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1 2   1 3   1 3   2 1   3 1   3 1   2 3   3 2   2 3   3 2
  3     2     3     3     2     3     1     1     3     3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1     3     3     2     3     3     3     3     3
  3     1     4     3     2     4     3     4     4
  4     4     1     4     4     2     4     3     4
.
  1
  2
  3
  4
  5

Mathematica

spsu[_, {}]:={{}}; spsu[foo_, set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsu[Select[foo, Complement[#, Complement[set, s]]=={}&], Complement[set, s]]]/@Cases[foo, {i, ___}];
ptnpos[y_]:=Position[Table[1, {#}]&/@y, 1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y], {k}], {k, Reverse[Union[y]]}], UnsameQ@@First/@#&];
Table[Sum[Length[Select[spsu[ptnverts[y], ptnpos[y]], Function[p, Sort[Length/@p]==Sort[y]]]], {y, IntegerPartitions[n]}], {n, 5}]

Crossrefs

Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.

Sequence in context: A125782 A343176 A307875 * A047143 A148779 A148780

Adjacent sequences: A321727 A321728 A321729 * A321731 A321732 A321733

Keyword

nonn,more

Author

Gus Wiseman, Nov 18 2018

Status

approved