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

0,5

Comments

A strict antichain is a finite set of finite nonempty sets, none of which is a subset of any other.

Links

Table of n, a(n) for n=0..11.

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

Cf. A000070, A000569, A006126, A096827, A209816, A283877, A293606, A293993, A306005, A318361, A319721, A321176, A321184.

Sequence in context: A368180 A071181 A213675 * A320322 A330106 A214795

Adjacent sequences: A321182 A321183 A321184 * A321186 A321187 A321188

Keyword

nonn,more

Author

Gus Wiseman, Oct 29 2018

Status

approved