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EXAMPLE
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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}}
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MATHEMATICA
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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}]
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