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A368186
Number of n-covers of an unlabeled n-set.
8
1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195
OFFSET
0,3
LINKS
FORMULA
a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
{{1}} {{1},{2}} {{1},{2},{3}}
{{1},{1,2}} {{1},{2},{1,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{2,3}}
{{1},{2},{1,2,3}}
{{1},{1,2},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1},{2,3},{1,2,3}}
{{1,2},{1,3},{1,2,3}}
MATHEMATICA
brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]], {n}], Union@@#==Range[n]&]]], {n, 0, 3}]
PROG
(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n, m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q, t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g, x, x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n, n) - G(n-1, n), n)) \\ Andrew Howroyd, Jan 03 2024
CROSSREFS
The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Sequence in context: A356615 A037172 A106163 * A359903 A360433 A278332
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 19 2023
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024
STATUS
approved