A368836 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on up to n vertices with k loops and n-k non-loops.

1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 2, 6, 6, 2, 1, 6, 17, 18, 8, 2, 1, 21, 52, 58, 30, 9, 2, 1, 65, 173, 191, 107, 37, 9, 2, 1, 221, 585, 666, 393, 148, 39, 9, 2, 1, 771, 2064, 2383, 1493, 589, 168, 40, 9, 2, 1, 2769, 7520, 8847, 5765, 2418, 718, 176, 40, 9, 2, 1

Offset: 0

Author

Gus Wiseman, Jan 11 2024

Keywords

• nonn
• tabl

Comments

Are the row sums the same as column k = 1 (shifted left)?

Yes. When k = 1 there is one loop. Remove the vertex with the loop and add loops to its neighbors. This process is reversible so there is a bijection. - Andrew Howroyd, Jan 13 2024

Examples

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  2  1
   2  6  6  2  1
   6 17 18  8  2  1
  21 52 58 30  9  2  1
Representatives of the loop-graphs counted by row n = 4:
  {12}{13}{14}{23} {1}{12}{13}{14} {1}{2}{12}{13} {1}{2}{3}{12} {1}{2}{3}{4}
  {12}{13}{24}{34} {1}{12}{13}{23} {1}{2}{12}{34} {1}{2}{3}{14}
                   {1}{12}{13}{24} {1}{2}{13}{14}
                   {1}{12}{23}{24} {1}{2}{13}{23}
                   {1}{12}{23}{34} {1}{2}{13}{24}
                   {1}{23}{24}{34} {1}{2}{13}{34}
		

Links

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

Crossrefs

Column k = 0 is A001434.

Row sums are A368598.

The labeled version is A368928.

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.

A058891 counts set-systems, unlabeled A000612.

Cf. A000272, A007717, A062740, A322661, A333331, A368596, A368730, A368835.

Programs

Mathematica

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k&]]], {n,0,4},{k,0,n}]

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}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
row(n) = {my(s=0, A=1+O(x*x^n)); forpart(p=n, s+=permcount(p) * polcoef(edges(p, i->A + x^i)*prod(i=1, #p, A + (x*y)^p[i]), n)); Vecrev(s/n!)} \\ Andrew Howroyd, Jan 13 2024

Extensions

a(28) onwards from Andrew Howroyd, Jan 13 2024