cp's OEIS Frontend

A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

%I A368095 #8 Dec 26 2023 08:33:49
%S A368095 1,1,2,4,8,17,39,86,208,508,1304
%N A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.
%C A368095 A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.
%C A368095 The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
%e A368095 Non-isomorphic representatives of the a(1) = 1 through a(5) = 17 set-systems:
%e A368095   {1}  {12}    {123}      {1234}        {12345}
%e A368095        {1}{2}  {1}{23}    {1}{234}      {1}{2345}
%e A368095                {2}{12}    {12}{34}      {12}{345}
%e A368095                {1}{2}{3}  {13}{23}      {14}{234}
%e A368095                           {3}{123}      {23}{123}
%e A368095                           {1}{2}{34}    {4}{1234}
%e A368095                           {1}{3}{23}    {1}{2}{345}
%e A368095                           {1}{2}{3}{4}  {1}{23}{45}
%e A368095                                         {1}{24}{34}
%e A368095                                         {1}{4}{234}
%e A368095                                         {2}{13}{23}
%e A368095                                         {2}{3}{123}
%e A368095                                         {3}{13}{23}
%e A368095                                         {4}{12}{34}
%e A368095                                         {1}{2}{3}{45}
%e A368095                                         {1}{2}{4}{34}
%e A368095                                         {1}{2}{3}{4}{5}
%t A368095 Table[Length[Select[bmp[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]], {n,0,10}]
%Y A368095 For labeled graphs we have A133686, complement A367867.
%Y A368095 For unlabeled graphs we have A134964, complement A140637.
%Y A368095 For set-systems we have A367902, complement A367903.
%Y A368095 These set-systems have BII-numbers A367906, complement A367907.
%Y A368095 The complement is A368094, connected A368409.
%Y A368095 Repeats allowed: A368098, ranks A368100, complement A368097, ranks A355529.
%Y A368095 Minimal multiset partitions not of this type are counted by A368187.
%Y A368095 The connected case is A368410.
%Y A368095 Factorizations of this type are counted by A368414, complement A368413.
%Y A368095 Allowing repeated edges gives A368422, complement A368421.
%Y A368095 A000110 counts set-partitions, non-isomorphic A000041.
%Y A368095 A003465 counts covering set-systems, unlabeled A055621.
%Y A368095 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368095 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368095 A283877 counts non-isomorphic set-systems, connected A300913.
%Y A368095 Cf. A001055, A007717, A302545, A306005, A316983, A319560, A321194, A321405, A330223, A367769, A367901.
%K A368095 nonn,more
%O A368095 0,3
%A A368095 _Gus Wiseman_, Dec 24 2023