A339659 - OEIS

%I #11 Jan 05 2021 19:07:39

%S 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,

%T 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,

%U 0,0,2,11,15,17,15,11,7,5,3,2,1,1

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

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

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

%e Triangle begins:

%e   1

%e   0 0 1

%e   0 0 0 1 1

%e   0 0 0 1 2 1 1

%e   0 0 0 0 2 3 2 1 1

%e   0 0 0 0 1 4 5 3 2 1 1

%e   0 0 0 0 1 4 7 7 5 3 2 1 1

%e For example, row n = 5 counts the following partitions:

%e   3322  22222  222211  2221111  22111111  211111111  1111111111

%e         32221  322111  3211111  31111111

%e         33211  331111  4111111

%e         42211  421111

%e                511111

%t prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Subsets[Union[m],{2}]}]];

%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t Table[Length[Select[strnorm[2*n],Length[Union[#]]==k&&Select[prpts[#],UnsameQ@@#&]!={}&]],{n,0,5},{k,0,2*n}]

%Y A000569 gives the row sums.

%Y A004250 is the central column.

%Y A005408 gives the row lengths.

%Y A008284/A072233 is the version counting all partitions.

%Y A259873 is the left half of the triangle.

%Y A309356 is a universal embedding.

%Y A027187 counts partitions of even length.

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

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

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

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

%Y - A000569 counts graphical partitions (A320922).

%Y - A058696 counts partitions of 2n (A300061).

%Y - A147878 counts connected multigraphical partitions (A320925).

%Y - A209816 counts multigraphical partitions (A320924).

%Y - A320921 counts connected graphical partitions (A320923).

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

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

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

%Y - A339656 counts loop-graphical partitions (A339658).

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

%K nonn,tabf

%O 0,14

%A _Gus Wiseman_, Dec 18 2020