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 A320921 Number of connected graphical partitions of 2n. 24
 1, 1, 1, 3, 5, 10, 19, 35, 60 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)