A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

1, 0, 1, 1, 3, 2, 7, 6, 15, 15, 30

Offset: 0

Author

Gus Wiseman, Oct 29 2018

Keywords

• nonn
• more

Examples

			The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (211)   (11111)  (222)     (3211)     (332)
               (1111)           (321)     (22111)    (422)
                                (2211)    (31111)    (431)
                                (3111)    (211111)   (2222)
                                (21111)   (1111111)  (3221)
                                (111111)             (3311)
                                                     (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(6) = 7 integer partitions together with a realizing multi-antichain of each (the parts of the partition count the appearances of each vertex in the multi-antichain):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2,3},{1,2,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}
		

Crossrefs

Cf. A000070, A000569, A006126, A096827, A147878, A209816, A283877, A318360, A319719, A319721, A320799, A320921, A321176.

Programs

Mathematica

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]]]];
multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]