OFFSET
0,4
COMMENTS
A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..11
Wikipedia, Degree (graph theory)
FORMULA
Binomial transform is A330056.
EXAMPLE
The a(3) = 5 set-systems:
{{1,2},{1,3},{2,3}}
{{1,2},{1,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
MATHEMATICA
Table[Length[Select[Subsets[Subsets[Range[n], {2, n}]], Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]], {n, 0, 4}]
PROG
(PARI) \\ here b(n) is A330056(n).
AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
b(n) = {sum(k=0, n, (-1)^k*binomial(n, k)*2^(2^(n-k)-(n-k)-1) * sum(j=0, k\2, sum(i=0, k-2*j, binomial(k, i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) )))}
a(n) = {sum(k=0, n, (-1)^k*binomial(n, k)*b(n-k))} \\ Andrew Howroyd, Jan 16 2023
CROSSREFS
The version for non-isomorphic set-systems is A330055 (by weight).
The non-covering version is A330056.
Set-systems with no singletons are A016031.
Set-systems with no endpoints are A330059.
Non-isomorphic set-systems with no singletons are A306005 (by weight).
Non-isomorphic set-systems with no endpoints are A330054 (by weight).
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 30 2019
EXTENSIONS
Terms a(5) and beyond from Andrew Howroyd, Jan 16 2023
STATUS
approved