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. These are a kind of partially labeled graphs.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 13 connected graphical necklaces.
Gus Wiseman, The a(5) = 148 connected graphical necklaces.
EXAMPLE
Inequivalent representatives of the a(2) = 1 through a(4) = 13 graphical necklaces:
{{12}} {{12}{13}} {{12}{13}{14}}
{{12}{13}{23}} {{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])];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], And[Union@@#==Range[n], Length[csm[#]]<=1, #=={}||#==First[Sort[Table[Nest[rotgra[#, n]&, #, j], {j, n}]]]]&]], {n, 0, 5}]
PROG
(PARI) \\ B(n, d) is graphs on n*d points invariant under 1/d rotation.
B(n, d)={2^(n*(n-1)*d/2 + n*(d\2))}
D(n, d)={my(v=vector(n, i, B(i, d)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); sumdiv(n, e, eulerphi(d*e) * moebius(e) * u[n/e] * e^(n/e-1))}
a(n)={if(n<=1, n==0, sumdiv(n, d, D(n/d, d))/n)} \\ Andrew Howroyd, Jan 24 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 01 2019
EXTENSIONS
Terms a(7) and beyond from Andrew Howroyd, Jan 24 2023
STATUS
approved