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
Examples
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}}
Crossrefs
Programs
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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}]
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