A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.
1, 0, 1, 1, 2, 2, 5, 5, 9, 11, 17, 20
Offset: 0
Examples
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}}
Crossrefs
Programs
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Mathematica
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}]
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