OFFSET
0,5
COMMENTS
A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.
EXAMPLE
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}}
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}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 29 2018
STATUS
approved