cp's OEIS Frontend

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

%I A321176 #14 Aug 01 2019 00:29:22
%S A321176 1,0,1,1,2,3,5,7,10,15,21,28
%N A321176 Number of integer partitions of n that are the vertex-degrees of some set system with no singletons.
%C A321176 A set system is a finite set of finite nonempty sets.
%e A321176 The a(2) = 1 through a(9) = 15 partitions:
%e A321176   (11)  (111)  (211)   (221)    (222)     (322)      (2222)      (333)
%e A321176                (1111)  (2111)   (2211)    (2221)     (3221)      (3222)
%e A321176                        (11111)  (3111)    (3211)     (3311)      (3321)
%e A321176                                 (21111)   (22111)    (22211)     (4221)
%e A321176                                 (111111)  (31111)    (32111)     (22221)
%e A321176                                           (211111)   (41111)     (32211)
%e A321176                                           (1111111)  (221111)    (33111)
%e A321176                                                      (311111)    (42111)
%e A321176                                                      (2111111)   (222111)
%e A321176                                                      (11111111)  (321111)
%e A321176                                                                  (411111)
%e A321176                                                                  (2211111)
%e A321176                                                                  (3111111)
%e A321176                                                                  (21111111)
%e A321176                                                                  (111111111)
%e A321176 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):
%e A321176      (41111): {{1,2},{1,3},{1,4},{1,5}}
%e A321176       (3311): {{1,2},{1,2,3},{1,2,4}}
%e A321176       (3221): {{1,2},{1,3},{1,2,3,4}}
%e A321176      (32111): {{1,2},{1,3},{1,2,4,5}}
%e A321176     (311111): {{1,2},{1,3},{1,4,5,6}}
%e A321176       (2222): {{1,2},{3,4},{1,2,3,4}}
%e A321176      (22211): {{1,2,3},{1,2,3,4,5}}
%e A321176     (221111): {{1,2},{1,2,3,4,5,6}}
%e A321176    (2111111): {{1,2},{1,3,4,5,6,7}}
%e A321176   (11111111): {{1,2,3,4,5,6,7,8}}
%t A321176 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A321176 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A321176 hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
%t A321176 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A321176 Table[Length[Select[strnorm[n],hyp[#]!={}&]],{n,8}]
%Y A321176 Cf. A000070, A000569, A147878, A209816, A283877, A306005, A318361, A320922, A320923, A320924, A321177.
%K A321176 nonn,more
%O A321176 0,5
%A A321176 _Gus Wiseman_, Oct 29 2018