A320921 Number of connected graphical partitions of 2n.

1, 1, 1, 3, 5, 10, 19, 35, 60

Offset: 0

Author

Gus Wiseman, Oct 24 2018

Keywords

• nonn
• more

Comments

An integer partition is connected and graphical if it comprises the multiset of vertex-degrees of some connected simple graph.

Examples

			The a(1) = 1 through a(6) = 19 connected graphical partitions:
  (11)  (211)  (222)   (2222)   (3322)    (3333)
               (2211)  (3221)   (22222)   (33222)
               (3111)  (22211)  (32221)   (33321)
                       (32111)  (33211)   (42222)
                       (41111)  (42211)   (43221)
                                (222211)  (222222)
                                (322111)  (322221)
                                (331111)  (332211)
                                (421111)  (333111)
                                (511111)  (422211)
                                          (432111)
                                          (522111)
                                          (2222211)
                                          (3222111)
                                          (3321111)
                                          (4221111)
                                          (4311111)
                                          (5211111)
                                          (6111111)
		

Links

• Gus Wiseman, Connected simple graphs realizing each of the a(6) = 19 connected graphical partitions of 12.

Crossrefs

Cf. A000070, A000569, A007717, A025065, A096373, A147878, A209816, A320891, A320911, A320923.

Programs

Mathematica

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[strnorm[2*n],Select[prptns[#],And[UnsameQ@@#,Length[csm[#]]==1]&]!={}&]],{n,5}]