A320921 Number of connected graphical partitions of 2n.

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

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..8.

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

Example

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)

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}]

Crossrefs

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

Sequence in context: A192860 A125750 A018168 * A084321 A323812 A270715

Adjacent sequences: A320918 A320919 A320920 * A320922 A320923 A320924

Keyword

nonn,more

Author

Gus Wiseman, Oct 24 2018

Status

approved