A321184 - OEIS

%I #7 Aug 01 2019 00:29:31

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

%N 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.

%e The a(2) = 1 through a(8) = 15 partitions:

%e   (11)  (111)  (22)    (2111)   (33)      (2221)     (44)

%e                (211)   (11111)  (222)     (3211)     (332)

%e                (1111)           (321)     (22111)    (422)

%e                                 (2211)    (31111)    (431)

%e                                 (3111)    (211111)   (2222)

%e                                 (21111)   (1111111)  (3221)

%e                                 (111111)             (3311)

%e                                                      (4211)

%e                                                      (22211)

%e                                                      (32111)

%e                                                      (41111)

%e                                                      (221111)

%e                                                      (311111)

%e                                                      (2111111)

%e                                                      (11111111)

%e 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):

%e       (33): {{1,2},{1,2},{1,2}}

%e      (321): {{1,2},{1,2},{1,3}}

%e     (3111): {{1,2},{1,3},{1,4}}

%e      (222): {{1,2,3},{1,2,3}}

%e     (2211): {{1,2,3},{1,2,4}}

%e    (21111): {{1,2},{1,3,4,5}}

%e   (111111): {{1,2,3,4,5,6}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];

%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]

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

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Oct 29 2018