OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 19 2023
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024
STATUS
approved