A330057 - OEIS

%I #9 Jan 16 2023 14:49:31

%S 1,0,0,5,1703,66954642,144115175199102143,

%T 1329227995784915808340204290157341181,

%U 226156424291633194186662080095093568664788471116325389572604136316742486364

%N Number of set-systems covering n vertices with no singletons or endpoints.

%C 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).

%H Andrew Howroyd, <a href="/A330057/b330057.txt">Table of n, a(n) for n = 0..11</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Degree_(graph_theory)">Degree (graph theory)</a>

%F Binomial transform is A330056.

%e The a(3) = 5 set-systems:

%e   {{1,2},{1,3},{2,3}}

%e   {{1,2},{1,3},{1,2,3}}

%e   {{1,2},{2,3},{1,2,3}}

%e   {{1,3},{2,3},{1,2,3}}

%e   {{1,2},{1,3},{2,3},{1,2,3}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{2,n}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]>1&]],{n,0,4}]

%o (PARI) \\ here b(n) is A330056(n).

%o AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}

%o 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)) )))}

%o a(n) = {sum(k=0, n, (-1)^k*binomial(n,k)*b(n-k))} \\ _Andrew Howroyd_, Jan 16 2023

%Y The version for non-isomorphic set-systems is A330055 (by weight).

%Y The non-covering version is A330056.

%Y Set-systems with no singletons are A016031.

%Y Set-systems with no endpoints are A330059.

%Y Non-isomorphic set-systems with no singletons are A306005 (by weight).

%Y Non-isomorphic set-systems with no endpoints are A330054 (by weight).

%Y Non-isomorphic set-systems counted by vertices are A000612.

%Y Non-isomorphic set-systems counted by weight are A283877.

%Y Cf. A007716, A055621, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 30 2019

%E Terms a(5) and beyond from _Andrew Howroyd_, Jan 16 2023