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
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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 24 2018
STATUS
approved