OFFSET
0,5
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
1
0 1
0 2 1
0 3 4 1
0 5 12 5 1
0 7 28 22 5 1
0 11 66 83 31 5 1
0 15 134 252 147 34 5 1
0 22 280 726 620 203 35 5 1
0 30 536 1946 2283 1069 235 35 5 1
0 42 1043 4982 7890 5019 1469 248 35 5 1
...
Row n = 4 counts the following representatives:
. {{1,1,1,1}} {{1},{1,1,1}} {{1},{2},{1,1}} {{1},{2},{3},{4}}
{{1,1,1,2}} {{1},{1,1,2}} {{1},{2},{1,2}}
{{1,1,2,2}} {{1},{1,2,2}} {{1},{2},{1,3}}
{{1,1,2,3}} {{1},{1,2,3}} {{1},{2},{3,3}}
{{1,2,3,4}} {{1},{2,2,2}} {{1},{2},{3,4}}
{{1},{2,2,3}}
{{1},{2,3,4}}
{{1,1},{1,2}}
{{1,1},{2,2}}
{{1,1},{2,3}}
{{1,2},{1,3}}
{{1,2},{3,4}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]], {s, Flatten[MapIndexed[Table[#2, {#1}]&, #]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i, p[[i]]}, {i, Length[p]}])], {p, Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], UnsameQ@@#&&Length[#]==k&]]], {n, 0, 5}, {k, 0, n}]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Dec 31 2023
STATUS
approved