cp's OEIS Frontend

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

%I A321185 #15 Jun 09 2021 02:32:33
%S A321185 1,0,1,1,2,2,5,5,9,11,17,20
%N A321185 Number of integer partitions of n that are the vertex-degrees of some strict antichain of sets with no singletons.
%C A321185 A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.
%e A321185 The a(2) = 1 through a(9) = 11 partitions:
%e A321185   (11)  (111)  (211)   (2111)   (222)     (2221)     (2222)      (3222)
%e A321185                (1111)  (11111)  (2211)    (22111)    (3221)      (22221)
%e A321185                                 (3111)    (31111)    (22211)     (32211)
%e A321185                                 (21111)   (211111)   (32111)     (33111)
%e A321185                                 (111111)  (1111111)  (41111)     (222111)
%e A321185                                                      (221111)    (321111)
%e A321185                                                      (311111)    (411111)
%e A321185                                                      (2111111)   (2211111)
%e A321185                                                      (11111111)  (3111111)
%e A321185                                                                  (21111111)
%e A321185                                                                  (111111111)
%e A321185 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):
%e A321185      (41111): {{1,2},{1,3},{1,4},{1,5}}
%e A321185       (3221): {{1,2},{1,3},{1,4},{2,3}}
%e A321185      (32111): {{1,3},{1,2,4},{1,2,5}}
%e A321185     (311111): {{1,2},{1,3},{1,4,5,6}}
%e A321185       (2222): {{1,2},{1,3,4},{2,3,4}}
%e A321185      (22211): {{1,2,3,4},{1,2,3,5}}
%e A321185     (221111): {{1,2,3},{1,2,4,5,6}}
%e A321185    (2111111): {{1,2},{1,3,4,5,6,7}}
%e A321185   (11111111): {{1,2,3,4,5,6,7,8}}
%t A321185 submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{___,x_,W___}}/;submultisetQ[{Z},{W}]]];
%t A321185 stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
%t A321185 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321185 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321185 anti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1,stableQ[#]]&];
%t A321185 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A321185 Table[Length[Select[strnorm[n],anti[#]!={}&]],{n,8}]
%Y A321185 Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.
%K A321185 nonn,more
%O A321185 0,5
%A A321185 _Gus Wiseman_, Oct 29 2018