A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 5, 3, 1, 1, 5, 12, 7, 3, 1, 1, 14, 29, 19, 8, 3, 1, 1, 35, 75, 47, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1

Offset: 0

Author

Gus Wiseman, Jan 13 2024

Keywords

• nonn
• tabl

Comments

Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.

Examples

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  1  1
   2  5  3  1  1
   5 12  7  3  1  1
  14 29 19  8  3  1  1
  35 75 47 21  8  3  1  1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

The case of a unique choice is A106234, row sums A000081.

Column k = 0 is A137917, labeled version A137916.

Without the choice condition we have A368836.

The labeled version is A368924, row sums maybe A333331.

Row sums are A368984, complement A368835.

A000085, A100861, A111924 count set partitions into singletons or pairs.

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A014068 counts loop-graphs, unlabeled A000666.

A322661 counts labeled covering half-loop-graphs, connected A062740.

Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Programs

Mathematica

Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]

PARI

\\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

Extensions

a(36) onwards from Andrew Howroyd, Jan 14 2024