A209816 -id:A209816 - cp's OEIS frontend

A328863 Number of partitions of 2n that describe the degree sequence of exactly one labeled multigraph with no loops.

1, 2, 4, 6, 9, 14, 19, 27, 37, 50, 66, 89, 115, 151, 195, 252, 321, 412, 520, 660, 829, 1042, 1299, 1623, 2010, 2492, 3071, 3783, 4635, 5679, 6922, 8434, 10234, 12406, 14985, 18085, 21751, 26135, 31312, 37471, 44723, 53321, 63415, 75336, 89303, 105734, 124938

Offset: 1

Peter Kagey, Oct 28 2019

nonn

Also the number of partitions of 2*n either with largest part equal to n or with three parts and largest part less than n.

			For n = 4, the a(4) = 6 partitions of 2*4 = 8 that describe a degree sequence of exactly one labeled multigraph are
  4 + 4,
  4 + 3 + 1,
  4 + 2 + 2,
  4 + 2 + 1 + 1,
  4 + 1 + 1 + 1 + 1, and
  3 + 3 + 2.

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000041, A069905, A209816.

a(n) = A000041(n) + A069905(n).

A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 9, 11, 17, 20

Offset: 0

Gus Wiseman, Oct 29 2018

A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (211)   (2111)   (222)     (2221)     (2222)      (3222)
               (1111)  (11111)  (2211)    (22111)    (3221)      (22221)
                                (3111)    (31111)    (22211)     (32211)
                                (21111)   (211111)   (32111)     (33111)
                                (111111)  (1111111)  (41111)     (222111)
                                                     (221111)    (321111)
                                                     (311111)    (411111)
                                                     (2111111)   (2211111)
                                                     (11111111)  (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 9 integer partitions together with a realizing strict antichain for each (the parts of the partition count the appearances of each vertex in the antichain):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3221): {{1,2},{1,3},{1,4},{2,3}}
     (32111): {{1,3},{1,2,4},{1,2,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{1,3,4},{2,3,4}}
     (22211): {{1,2,3,4},{1,2,3,5}}
    (221111): {{1,2,3},{1,2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
anti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],anti[#]!={}&]],{n,8}]

cp's OEIS Frontend

A328863 Number of partitions of 2n that describe the degree sequence of exactly one labeled multigraph with no loops.

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