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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

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

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