OFFSET
0,14
COMMENTS
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.
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:
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Dec 18 2020
STATUS
approved