A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

1, 0, 1, 0, 2, 1, 1, 9, 9, 1, 15, 80, 90, 24, 1, 252, 1050, 1200, 450, 50, 1, 5005, 18018, 20475, 9100, 1575, 90, 1, 116280, 379848, 427329, 209475, 46550, 4410, 147, 1, 3108105, 9472320, 10548720, 5503680, 1433250, 183456, 10584, 224, 1

Offset: 0

Author

Gus Wiseman, Jan 11 2024

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1
     0     1
     0     2     1
     1     9     9     1
    15    80    90    24     1
   252  1050  1200   450    50     1
  5005 18018 20475  9100  1575    90     1
The loop-graphs counted in row n = 3 (loops shown as singletons):
  {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}
                {1}{12}{23}  {1}{2}{13}
                {1}{13}{23}  {1}{2}{23}
                {2}{12}{13}  {1}{3}{12}
                {2}{12}{23}  {1}{3}{13}
                {2}{13}{23}  {1}{3}{23}
                {3}{12}{13}  {2}{3}{12}
                {3}{12}{23}  {2}{3}{13}
                {3}{13}{23}  {2}{3}{23}
		

Links

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

Crossrefs

Row sums are A014068, unlabeled version A000666.

Column k = 0 is A116508, covering version A367863.

The covering case is A368597.

The unlabeled version is A368836.

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

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A058891 counts set-systems (without singletons A016031), unlabeled A000612.

A322661 counts labeled covering loop-graphs, connected A062740.

Cf. A057500, A079491, A339065, A368596, A368927.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]
T[n_,k_]:= Binomial[n,k]*Binomial[Binomial[n,2],n-k]; Table[T[n,k],{n,0,8},{k,0,n}]// Flatten (* Stefano Spezia, Jan 14 2024 *)

PARI

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ Andrew Howroyd, Jan 14 2024

Formula

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k).