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A339659
Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.
16
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.
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).
Sequence in context: A057276 A259829 A035185 * A338971 A244600 A288558
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Dec 18 2020
STATUS
approved