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A327227 Number of labeled simple graphs covering n vertices with at least one endpoint/leaf. 22

%I

%S 0,0,1,3,31,515,15381,834491,83016613,15330074139,5324658838645,

%T 3522941267488973,4489497643961740521,11119309286377621015089,

%U 53893949089393110881259181,513788884660608277842596504415,9669175277199248753133328740702449

%N Number of labeled simple graphs covering n vertices with at least one endpoint/leaf.

%C Covering means there are no isolated vertices.

%C A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

%C Also graphs with minimum vertex-degree 1.

%H Andrew Howroyd, <a href="/A327227/b327227.txt">Table of n, a(n) for n = 0..50</a>

%F Inverse binomial transform of A245797, if we assume A245797(0) = 0.

%e The a(4) = 31 edge-sets:

%e {12,34} {12,13,14} {12,13,14,23}

%e {13,24} {12,13,24} {12,13,14,24}

%e {14,23} {12,13,34} {12,13,14,34}

%e {12,14,23} {12,13,23,24}

%e {12,14,34} {12,13,23,34}

%e {12,23,24} {12,14,23,24}

%e {12,23,34} {12,14,24,34}

%e {12,24,34} {12,23,24,34}

%e {13,14,23} {13,14,23,34}

%e {13,14,24} {13,14,24,34}

%e {13,23,24} {13,23,24,34}

%e {13,23,34} {14,23,24,34}

%e {13,24,34}

%e {14,23,24}

%e {14,23,34}

%e {14,24,34}

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

%Y Column k=1 of A327366.

%Y The non-covering version is A245797.

%Y The unlabeled version is A324693.

%Y The generalization to set-systems is A327229.

%Y BII-numbers of set-systems with minimum degree 1 are A327105.

%Y Cf. A001187, A006129, A059166, A059167, A100743, A136284, A327079, A327098, A327103, A327228, A327230.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 01 2019

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Last modified March 3 09:37 EST 2021. Contains 341760 sequences. (Running on oeis4.)