A321176 Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.

1, 0, 1, 1, 2, 3, 5, 7, 10, 15, 21, 28

Offset

0,5

Comments

A set system is a finite set of finite nonempty sets.

Links

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

Example

The a(2) = 1 through a(9) = 15 partitions:
  (11)  (111)  (211)   (221)    (222)     (322)      (2222)      (333)
               (1111)  (2111)   (2211)    (2221)     (3221)      (3222)
                       (11111)  (3111)    (3211)     (3311)      (3321)
                                (21111)   (22111)    (22211)     (4221)
                                (111111)  (31111)    (32111)     (22221)
                                          (211111)   (41111)     (32211)
                                          (1111111)  (221111)    (33111)
                                                     (311111)    (42111)
                                                     (2111111)   (222111)
                                                     (11111111)  (321111)
                                                                 (411111)
                                                                 (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)
The a(8) = 10 integer partitions together with a realizing set system for each (the parts of the partition count the appearances of each vertex in the set system):
     (41111): {{1,2},{1,3},{1,4},{1,5}}
      (3311): {{1,2},{1,2,3},{1,2,4}}
      (3221): {{1,2},{1,3},{1,2,3,4}}
     (32111): {{1,2},{1,3},{1,2,4,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
      (2222): {{1,2},{3,4},{1,2,3,4}}
     (22211): {{1,2,3},{1,2,3,4,5}}
    (221111): {{1,2},{1,2,3,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}

Mathematica

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]]]];
hyp[m_]:=Select[mps[m], And[And@@UnsameQ@@@#, UnsameQ@@#, Min@@Length/@#>1]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n], hyp[#]!={}&]], {n, 8}]

Crossrefs

Cf. A000070, A000569, A147878, A209816, A283877, A306005, A318361, A320922, A320923, A320924, A321177.

Sequence in context: A011972 A338553 A272402 * A240573 A160571 A238543

Adjacent sequences: A321173 A321174 A321175 * A321177 A321178 A321179

Keyword

nonn,more

Author

Gus Wiseman, Oct 29 2018

Status

approved