cp's OEIS Frontend

A320921 Number of connected graphical partitions of 2n.

%I A320921 #9 Nov 14 2018 11:30:16
%S A320921 1,1,1,3,5,10,19,35,60
%N A320921 Number of connected graphical partitions of 2n.
%C A320921 An integer partition is connected and graphical if it comprises the multiset of vertex-degrees of some connected simple graph.
%H A320921 Gus Wiseman, <a href="/A320921/a320921.png">Connected simple graphs realizing each of the a(6) = 19 connected graphical partitions of 12.</a>
%e A320921 The a(1) = 1 through a(6) = 19 connected graphical partitions:
%e A320921   (11)  (211)  (222)   (2222)   (3322)    (3333)
%e A320921                (2211)  (3221)   (22222)   (33222)
%e A320921                (3111)  (22211)  (32221)   (33321)
%e A320921                        (32111)  (33211)   (42222)
%e A320921                        (41111)  (42211)   (43221)
%e A320921                                 (222211)  (222222)
%e A320921                                 (322111)  (322221)
%e A320921                                 (331111)  (332211)
%e A320921                                 (421111)  (333111)
%e A320921                                 (511111)  (422211)
%e A320921                                           (432111)
%e A320921                                           (522111)
%e A320921                                           (2222211)
%e A320921                                           (3222111)
%e A320921                                           (3321111)
%e A320921                                           (4221111)
%e A320921                                           (4311111)
%e A320921                                           (5211111)
%e A320921                                           (6111111)
%t A320921 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[[#]]&]}]]];
%t A320921 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A320921 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]]]]]]]]];
%t A320921 Table[Length[Select[strnorm[2*n],Select[prptns[#],And[UnsameQ@@#,Length[csm[#]]==1]&]!={}&]],{n,5}]
%Y A320921 Cf. A000070, A000569, A007717, A025065, A096373, A147878, A209816, A320891, A320911, A320923.
%K A320921 nonn,more
%O A320921 0,4
%A A320921 _Gus Wiseman_, Oct 24 2018