A330056 Number of set-systems with n vertices and no singletons or endpoints.

1, 1, 1, 6, 1724, 66963208, 144115175600855641, 1329227995784915809349010517957163445, 226156424291633194186662080095093568675422295082604716043360995547325655259

Offset: 0

Author

Gus Wiseman, Nov 30 2019

Keywords

• nonn

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).

Examples

			The a(3) = 6 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}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..11
• Wikipedia, Degree (graph theory)

Crossrefs

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

The covering case is A330057.

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.

Cf. A007716, A055621, A008299, A302545, A317533, A317794, A319559, A320665, A321405, A330052, A330058.

Programs

Mathematica

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

PARI

\\ Here AS2(n,k) is A008299 (associated Stirling of 2nd kind)
AS2(n, k) = {sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) )}
a(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)) )))} \\ Andrew Howroyd, Jan 16 2023

Formula

Binomial transform of A330057.

a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} Sum_{i=0..k-2*j} (-1)^k * binomial(n,k) * 2^(2^(n-k)-(n-k)-1) * binomial(k,i) * AS2(k-i, j) * (2^(n-k)-1)^i * 2^(j*(n-k)) where AS2(n,k) are the associated Stirling numbers of the 2nd kind (A008299). - Andrew Howroyd, Jan 16 2023

Extensions

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