cp's OEIS Frontend

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

%I A339659 #11 Jan 05 2021 19:07:39
%S A339659 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 A339659 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 A339659 0,0,2,11,15,17,15,11,7,5,3,2,1,1
%N A339659 Irregular triangle read by rows where T(n,k) is the number of graphical partitions of 2n into k parts, 0 <= k <= 2n.
%C A339659 Conjecture: The column sums 1, 0, 1, 2, 7, 20, 67, ... are given by A304787.
%C A339659 An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.
%e A339659 Triangle begins:
%e A339659   1
%e A339659   0 0 1
%e A339659   0 0 0 1 1
%e A339659   0 0 0 1 2 1 1
%e A339659   0 0 0 0 2 3 2 1 1
%e A339659   0 0 0 0 1 4 5 3 2 1 1
%e A339659   0 0 0 0 1 4 7 7 5 3 2 1 1
%e A339659 For example, row n = 5 counts the following partitions:
%e A339659   3322  22222  222211  2221111  22111111  211111111  1111111111
%e A339659         32221  322111  3211111  31111111
%e A339659         33211  331111  4111111
%e A339659         42211  421111
%e A339659                511111
%t A339659 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 A339659 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A339659 Table[Length[Select[strnorm[2*n],Length[Union[#]]==k&&Select[prpts[#],UnsameQ@@#&]!={}&]],{n,0,5},{k,0,2*n}]
%Y A339659 A000569 gives the row sums.
%Y A339659 A004250 is the central column.
%Y A339659 A005408 gives the row lengths.
%Y A339659 A008284/A072233 is the version counting all partitions.
%Y A339659 A259873 is the left half of the triangle.
%Y A339659 A309356 is a universal embedding.
%Y A339659 A027187 counts partitions of even length.
%Y A339659 A339559 = partitions that cannot be partitioned into distinct strict pairs.
%Y A339659 A339560 = partitions that can be partitioned into distinct strict pairs.
%Y A339659 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339659 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339659 - A000569 counts graphical partitions (A320922).
%Y A339659 - A058696 counts partitions of 2n (A300061).
%Y A339659 - A147878 counts connected multigraphical partitions (A320925).
%Y A339659 - A209816 counts multigraphical partitions (A320924).
%Y A339659 - A320921 counts connected graphical partitions (A320923).
%Y A339659 - A321728 is conjectured to count non-half-loop-graphical partitions of n.
%Y A339659 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339659 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339659 - A339656 counts loop-graphical partitions (A339658).
%Y A339659 Cf. A000219, A002100, A006881, A007717, A025065, A320656, A320894, A338914, A338916, A339561, A339661.
%K A339659 nonn,tabf
%O A339659 0,14
%A A339659 _Gus Wiseman_, Dec 18 2020