The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 20 05:04 EDT 2021. Contains 347577 sequences. (Running on oeis4.)