cp's OEIS Frontend

A330059 Number of set-systems with n vertices and no endpoints.

%I A330059 #19 Dec 18 2024 23:31:45
%S A330059 1,1,2,63,29471,2144945976,9223371624669871587,
%T A330059 170141183460469227599616678821978424151,
%U A330059 57896044618658097711785492504343953752410420469299789800819363538011879603532
%N A330059 Number of set-systems with n vertices and no endpoints.
%C A330059 A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1).
%H A330059 Andrew Howroyd, <a href="/A330059/b330059.txt">Table of n, a(n) for n = 0..11</a>
%F A330059 a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^k * binomial(n,k) * 2^(2^(n-k)-1) * Stirling2(k,j) * 2^(j*(n-k)). - _Andrew Howroyd_, Jan 16 2023
%e A330059 The a(2) = 2 set-systems are {} and {{1},{2},{1,2}}. The a(3) = 63 set-systems are:
%e A330059   0                 {2}{3}{12}{13}       {1}{3}{12}{13}{23}
%e A330059   {1}{2}{12}        {2}{12}{13}{23}      {2}{3}{12}{13}{23}
%e A330059   {1}{3}{13}        {2}{3}{12}{123}      {1}{2}{12}{23}{123}
%e A330059   {2}{3}{23}        {2}{3}{13}{123}      {1}{2}{13}{23}{123}
%e A330059   {12}{13}{23}      {3}{12}{13}{23}      {1}{3}{12}{13}{123}
%e A330059   {1}{23}{123}      {1}{13}{23}{123}     {1}{3}{12}{23}{123}
%e A330059   {2}{13}{123}      {2}{12}{13}{123}     {1}{3}{13}{23}{123}
%e A330059   {3}{12}{123}      {2}{12}{23}{123}     {2}{3}{12}{13}{123}
%e A330059   {12}{13}{123}     {2}{13}{23}{123}     {2}{3}{12}{23}{123}
%e A330059   {12}{23}{123}     {3}{12}{13}{123}     {2}{3}{13}{23}{123}
%e A330059   {13}{23}{123}     {3}{12}{23}{123}     {1}{12}{13}{23}{123}
%e A330059   {1}{2}{13}{23}    {3}{13}{23}{123}     {2}{12}{13}{23}{123}
%e A330059   {1}{2}{3}{123}    {12}{13}{23}{123}    {3}{12}{13}{23}{123}
%e A330059   {1}{3}{12}{23}    {1}{2}{3}{12}{13}    {1}{2}{3}{12}{13}{23}
%e A330059   {1}{12}{13}{23}   {1}{2}{3}{12}{23}    {1}{2}{3}{12}{13}{123}
%e A330059   {1}{2}{13}{123}   {1}{2}{3}{13}{23}    {1}{2}{3}{12}{23}{123}
%e A330059   {1}{2}{23}{123}   {1}{2}{12}{13}{23}   {1}{2}{3}{13}{23}{123}
%e A330059   {1}{3}{12}{123}   {1}{2}{3}{12}{123}   {1}{2}{12}{13}{23}{123}
%e A330059   {1}{3}{23}{123}   {1}{2}{3}{13}{123}   {1}{3}{12}{13}{23}{123}
%e A330059   {1}{12}{13}{123}  {1}{2}{3}{23}{123}   {2}{3}{12}{13}{23}{123}
%e A330059   {1}{12}{23}{123}  {1}{2}{12}{13}{123}  {1}{2}{3}{12}{13}{23}{123}
%t A330059 Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]
%o A330059 (PARI) a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*2^(2^(n-k)-1)*sum(j=0, k, stirling(k,j,2)*2^(j*(n-k)) ))} \\ _Andrew Howroyd_, Jan 16 2023
%Y A330059 The case with no singletons is A330056.
%Y A330059 The unlabeled version is A330054 (by weight) or A330124 (by vertices).
%Y A330059 Set-systems with no singletons are A016031.
%Y A330059 Non-isomorphic set-systems with no singletons are A306005 (by weight).
%Y A330059 Cf. A000612, A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330057, A330058.
%K A330059 nonn
%O A330059 0,3
%A A330059 _Gus Wiseman_, Dec 01 2019
%E A330059 Terms a(5) and beyond from _Andrew Howroyd_, Jan 16 2023