OFFSET
0,3
COMMENTS
A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..11
FORMULA
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
EXAMPLE
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}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Min@@Length/@Split[Sort[Join@@#]]>1&]], {n, 0, 4}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 01 2019
EXTENSIONS
Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023
STATUS
approved