A330124 -id:A330124 - cp's OEIS frontend

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

1, 1, 2, 63, 29471, 2144945976, 9223371624669871587, 170141183460469227599616678821978424151, 57896044618658097711785492504343953752410420469299789800819363538011879603532

Offset: 0

Gus Wiseman, Dec 01 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1).

			The a(2) = 2 set-systems are {} and {{1},{2},{1,2}}. The a(3) = 63 set-systems are:
  0                 {2}{3}{12}{13}       {1}{3}{12}{13}{23}
  {1}{2}{12}        {2}{12}{13}{23}      {2}{3}{12}{13}{23}
  {1}{3}{13}        {2}{3}{12}{123}      {1}{2}{12}{23}{123}
  {2}{3}{23}        {2}{3}{13}{123}      {1}{2}{13}{23}{123}
  {12}{13}{23}      {3}{12}{13}{23}      {1}{3}{12}{13}{123}
  {1}{23}{123}      {1}{13}{23}{123}     {1}{3}{12}{23}{123}
  {2}{13}{123}      {2}{12}{13}{123}     {1}{3}{13}{23}{123}
  {3}{12}{123}      {2}{12}{23}{123}     {2}{3}{12}{13}{123}
  {12}{13}{123}     {2}{13}{23}{123}     {2}{3}{12}{23}{123}
  {12}{23}{123}     {3}{12}{13}{123}     {2}{3}{13}{23}{123}
  {13}{23}{123}     {3}{12}{23}{123}     {1}{12}{13}{23}{123}
  {1}{2}{13}{23}    {3}{13}{23}{123}     {2}{12}{13}{23}{123}
  {1}{2}{3}{123}    {12}{13}{23}{123}    {3}{12}{13}{23}{123}
  {1}{3}{12}{23}    {1}{2}{3}{12}{13}    {1}{2}{3}{12}{13}{23}
  {1}{12}{13}{23}   {1}{2}{3}{12}{23}    {1}{2}{3}{12}{13}{123}
  {1}{2}{13}{123}   {1}{2}{3}{13}{23}    {1}{2}{3}{12}{23}{123}
  {1}{2}{23}{123}   {1}{2}{12}{13}{23}   {1}{2}{3}{13}{23}{123}
  {1}{3}{12}{123}   {1}{2}{3}{12}{123}   {1}{2}{12}{13}{23}{123}
  {1}{3}{23}{123}   {1}{2}{3}{13}{123}   {1}{3}{12}{13}{23}{123}
  {1}{12}{13}{123}  {1}{2}{3}{23}{123}   {2}{3}{12}{13}{23}{123}
  {1}{12}{23}{123}  {1}{2}{12}{13}{123}  {1}{2}{3}{12}{13}{23}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..11

The case with no singletons is A330056.
The unlabeled version is A330054 (by weight) or A330124 (by vertices).
Set-systems with no singletons are A016031.
Non-isomorphic set-systems with no singletons are A306005 (by weight).
Cf. A000612, A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330057, A330058.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]

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

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

Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023

A330196 Number of unlabeled set-systems covering n vertices with no endpoints.

Original entry on oeis.org

1, 0, 1, 20, 1754

Offset: 0

Gus Wiseman, Dec 05 2019

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

			Non-isomorphic representatives of the a(3) = 20 set-systems:
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

First differences of the non-covering version A330124.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by vertices are A317794.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
Cf. A007716, A016031, A306005, A317795, A320665, A330052, A330053, A330055.

cp's OEIS Frontend

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

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