A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

1, 1, 3, 9, 37, 152, 780, 3965, 23460, 141471, 944217, 6445643, 48075092, 364921557, 2974423953, 24847873439, 219611194148, 1987556951714, 18930298888792, 184244039718755, 1874490999743203, 19510832177784098, 210941659716920257, 2331530519337226199, 26692555830628617358

Offset

0,3

Comments

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:

1 2 3

1 2

2 3

Links

Table of n, a(n) for n=0..24.

Example

The a(4) = 37 partitions into vertical sections of integer partitions of 4:
  1 2 3 4
.
  1 2 3   1 2 3   1 2 3   1 2 3
  4       3       2       1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3 4   2 3   3 2   1 3   1 2   3 1   2 1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1
.
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
  2   2   2   2   2   1   1   2   2   2   2   1   1   2   1
  3   3   2   3   2   2   2   1   1   3   2   1   2   1   1
  4   3   3   2   2   3   2   3   2   1   1   2   1   1   1

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[Rest[Subsets[ptnpos[y]]], UnsameQ@@First/@#&];
Table[Sum[Length[spsu[ptnverts[y], ptnpos[y]]], {y, IntegerPartitions[n]}], {n, 6}]

Crossrefs

Cf. A000110, A000258, A008277, A046682, A122111, A318396, A321728, A321729, A321730, A321731, A321738, A321854.

Sequence in context: A149021 A149022 A134818 * A002751 A358397 A245890

Adjacent sequences: A321734 A321735 A321736 * A321738 A321739 A321740

Keyword

nonn

Author

Gus Wiseman, Nov 19 2018

Extensions

a(11)-a(24) from Ludovic Schwob, Aug 28 2023

Status

approved