OFFSET
0,4
COMMENTS
A graphical necklace is a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. Covering means there are no isolated vertices. These are a kind of partially labeled graphs.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(5) = 156 covering graphical necklaces.
FORMULA
a(n) = (1/n)*Sum{d|n} phi(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ). - Andrew Howroyd, Aug 19 2019
EXAMPLE
Inequivalent representatives of the a(2) = 1 through a(4) = 15 graphical necklaces:
{{12}} {{12}{13}} {{12}{34}}
{{12}{13}{23}} {{13}{24}}
{{12}{13}{14}}
{{12}{13}{24}}
{{12}{13}{34}}
{{12}{14}{23}}
{{12}{24}{34}}
{{12}{13}{14}{23}}
{{12}{13}{14}{24}}
{{12}{13}{14}{34}}
{{12}{13}{24}{34}}
{{12}{14}{23}{34}}
{{12}{13}{14}{23}{24}}
{{12}{13}{14}{23}{34}}
{{12}{13}{14}{23}{24}{34}}
MATHEMATICA
rotgra[g_, m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m, 1, k+1])];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], And[Union@@#==Range[n], #=={}||#==First[Sort[Table[Nest[rotgra[#, n]&, #, j], {j, n}]]]]&]], {n, 0, 5}]
PROG
(PARI) a(n)={if(n<1, n==0, sumdiv(n, d, eulerphi(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d, k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2))))/n)} \\ Andrew Howroyd, Aug 19 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 28 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019
STATUS
approved