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A330052 Number of non-isomorphic set-systems of weight n with at least one endpoint.

%I A330052 #9 Jan 27 2024 16:06:01
%S A330052 0,1,2,4,8,18,40,94,228,579,1508,4092,11478,33337,100016,309916,
%T A330052 990008,3257196,11021851,38314009,136657181,499570867,1869792499,
%U A330052 7158070137,28003286261,111857491266,455852284867,1893959499405,8017007560487,34552315237016,151534813272661
%N A330052 Number of non-isomorphic set-systems of weight n with at least one endpoint.
%C A330052 A set-system is a finite set of finite nonempty sets of positive integers. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H A330052 Andrew Howroyd, <a href="/A330052/b330052.txt">Table of n, a(n) for n = 0..50</a>
%H A330052 Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>
%F A330052 a(n) = A283877(n) - A330054(n). - _Andrew Howroyd_, Jan 27 2024
%e A330052 Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions:
%e A330052   {1}  {12}    {123}      {1234}        {12345}
%e A330052        {1}{2}  {1}{12}    {1}{123}      {1}{1234}
%e A330052                {1}{23}    {12}{13}      {12}{123}
%e A330052                {1}{2}{3}  {1}{234}      {12}{134}
%e A330052                           {12}{34}      {1}{2345}
%e A330052                           {1}{2}{13}    {12}{345}
%e A330052                           {1}{2}{34}    {1}{12}{13}
%e A330052                           {1}{2}{3}{4}  {1}{12}{23}
%e A330052                                         {1}{12}{34}
%e A330052                                         {1}{2}{123}
%e A330052                                         {1}{2}{134}
%e A330052                                         {1}{2}{345}
%e A330052                                         {1}{23}{45}
%e A330052                                         {2}{13}{14}
%e A330052                                         {1}{2}{3}{12}
%e A330052                                         {1}{2}{3}{14}
%e A330052                                         {1}{2}{3}{45}
%e A330052                                         {1}{2}{3}{4}{5}
%t A330052 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330052 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330052 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A330052 brute[{}]:={};brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
%t A330052 Table[Length[Select[Union[brute/@Join@@mps/@strnorm[n]],UnsameQ@@#&&And@@UnsameQ@@@#&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]
%Y A330052 The complement is counted by A330054.
%Y A330052 The multiset partition version is A330058.
%Y A330052 Non-isomorphic set-systems with at least one singleton are A330053.
%Y A330052 Non-isomorphic set-systems counted by vertices are A000612.
%Y A330052 Non-isomorphic set-systems counted by weight are A283877.
%Y A330052 Cf. A007716, A055621, A306005, A317533, A317794, A319559, A320665, A330055, A330056.
%K A330052 nonn
%O A330052 0,3
%A A330052 _Gus Wiseman_, Nov 30 2019
%E A330052 a(11) onwards from _Andrew Howroyd_, Jan 27 2024