A368096 Triangle read by rows where T(n,k) is the number of non-isomorphic set-systems of length k and weight n.
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 5, 8, 3, 1, 0, 1, 8, 18, 13, 3, 1, 0, 1, 9, 32, 37, 15, 3, 1, 0, 1, 13, 55, 96, 59, 16, 3, 1, 0, 1, 14, 91, 209, 196, 74, 16, 3, 1, 0, 1, 19, 138, 449, 573, 313, 82, 16, 3, 1, 0, 1, 20, 206, 863, 1529, 1147, 403, 84, 16, 3, 1
Offset: 0
Examples
Triangle begins:
1
0 1
0 1 1
0 1 2 1
0 1 4 3 1
0 1 5 8 3 1
0 1 8 18 13 3 1
0 1 9 32 37 15 3 1
0 1 13 55 96 59 16 3 1
0 1 14 91 209 196 74 16 3 1
0 1 19 138 449 573 313 82 16 3 1
...
Non-isomorphic representatives of the set-systems counted in row n = 5:
. {12345} {1}{1234} {1}{2}{123} {1}{2}{3}{12} {1}{2}{3}{4}{5}
{1}{2345} {1}{2}{134} {1}{2}{3}{14}
{12}{123} {1}{2}{345} {1}{2}{3}{45}
{12}{134} {1}{12}{13}
{12}{345} {1}{12}{23}
{1}{12}{34}
{1}{23}{24}
{1}{23}{45}
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Crossrefs
Programs
-
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@@#&&And@@UnsameQ@@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}] -
PARI
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={WeighT(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
Extensions
Terms a(66) and beyond from Andrew Howroyd, Jan 11 2024
Comments