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A368186 Number of n-covers of an unlabeled n-set.

%I A368186 #13 Jan 03 2024 15:11:15
%S A368186 1,1,2,9,87,1973,118827,20576251,10810818595,17821875542809,
%T A368186 94589477627232498,1651805220868992729874,96651473179540769701281003,
%U A368186 19238331716776641088273777321428,13192673305726630096303157068241728202,31503323006770789288222386469635474844616195
%N A368186 Number of n-covers of an unlabeled n-set.
%H A368186 Andrew Howroyd, <a href="/A368186/b368186.txt">Table of n, a(n) for n = 0..50</a>
%F A368186 a(n) = A055130(n, n) for n > 0. - _Andrew Howroyd_, Jan 03 2024
%e A368186 Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
%e A368186   {{1}}  {{1},{2}}    {{1},{2},{3}}
%e A368186          {{1},{1,2}}  {{1},{2},{1,3}}
%e A368186                       {{1},{1,2},{1,3}}
%e A368186                       {{1},{1,2},{2,3}}
%e A368186                       {{1},{2},{1,2,3}}
%e A368186                       {{1},{1,2},{1,2,3}}
%e A368186                       {{1,2},{1,3},{2,3}}
%e A368186                       {{1},{2,3},{1,2,3}}
%e A368186                       {{1,2},{1,3},{1,2,3}}
%t A368186 brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
%t A368186 Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]
%o A368186 (PARI)
%o A368186 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}
%o A368186 K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
%o A368186 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!)}
%o A368186 a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ _Andrew Howroyd_, Jan 03 2024
%Y A368186 The labeled version is A054780, ranks A367917, non-covering A367916.
%Y A368186 The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
%Y A368186 The case of connected graphs is A001429, labeled A057500.
%Y A368186 Covers with any number of edges are counted by A003465, unlabeled A055621.
%Y A368186 A046165 counts minimal covers, ranks A309326.
%Y A368186 A058891 counts set-systems, unlabeled A000612, without singletons A016031.
%Y A368186 A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
%Y A368186 Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.
%K A368186 nonn
%O A368186 0,3
%A A368186 _Gus Wiseman_, Dec 19 2023
%E A368186 Terms a(6) and beyond from _Andrew Howroyd_, Jan 03 2024