A339659 Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 2, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 5, 3, 2, 1, 1, 0, 0, 0, 0, 1, 4, 7, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 4, 9, 11, 11, 7, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0, 2, 11, 15, 17, 15, 11, 7, 5, 3, 2, 1, 1

Offset

0,14

Comments

Conjecture: The column sums 1, 0, 1, 2, 7, 20, 67, ... are given by A304787.

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.

Links

Table of n, a(n) for n=0..80.

Example

Triangle begins:
  1
  0 0 1
  0 0 0 1 1
  0 0 0 1 2 1 1
  0 0 0 0 2 3 2 1 1
  0 0 0 0 1 4 5 3 2 1 1
  0 0 0 0 1 4 7 7 5 3 2 1 1
For example, row n = 5 counts the following partitions:
  3322  22222  222211  2221111  22111111  211111111  1111111111
        32221  322111  3211111  31111111
        33211  331111  4111111
        42211  421111
               511111

Mathematica

prpts[m_]:=If[Length[m]==0, {{}}, Join@@Table[Prepend[#, ipr]&/@prpts[Fold[DeleteCases[#1, #2, {1}, 1]&, m, ipr]], {ipr, Subsets[Union[m], {2}]}]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n], Length[Union[#]]==k&&Select[prpts[#], UnsameQ@@#&]!={}&]], {n, 0, 5}, {k, 0, 2*n}]

Crossrefs

A000569 gives the row sums.

A004250 is the central column.

A005408 gives the row lengths.

A008284/A072233 is the version counting all partitions.

A259873 is the left half of the triangle.

A309356 is a universal embedding.

A027187 counts partitions of even length.

A339559 = partitions that cannot be partitioned into distinct strict pairs.

A339560 = partitions that can be partitioned into distinct strict pairs.

The following count vertex-degree partitions and give their Heinz numbers:

- A000070 counts non-multigraphical partitions of 2n (A339620).

- A000569 counts graphical partitions (A320922).

- A058696 counts partitions of 2n (A300061).

- A147878 counts connected multigraphical partitions (A320925).

- A209816 counts multigraphical partitions (A320924).

- A320921 counts connected graphical partitions (A320923).

- A321728 is conjectured to count non-half-loop-graphical partitions of n.

- A339617 counts non-graphical partitions of 2n (A339618).

- A339655 counts non-loop-graphical partitions of 2n (A339657).

- A339656 counts loop-graphical partitions (A339658).

Cf. A000219, A002100, A006881, A007717, A025065, A320656, A320894, A338914, A338916, A339561, A339661.

Sequence in context: A057276 A259829 A035185 * A338971 A244600 A288558

Adjacent sequences: A339656 A339657 A339658 * A339660 A339661 A339662

Keyword

nonn,tabf

Author

Gus Wiseman, Dec 18 2020

Status

approved