A321730 - OEIS

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

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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 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

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