A368928 - OEIS

%I #18 Jan 14 2024 16:11:04

%S 1,0,1,0,2,1,1,9,9,1,15,80,90,24,1,252,1050,1200,450,50,1,5005,18018,

%T 20475,9100,1575,90,1,116280,379848,427329,209475,46550,4410,147,1,

%U 3108105,9472320,10548720,5503680,1433250,183456,10584,224,1

%N 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.

%H Andrew Howroyd, <a href="/A368928/b368928.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

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

%e Triangle begins:

%e      1

%e      0     1

%e      0     2     1

%e      1     9     9     1

%e     15    80    90    24     1

%e    252  1050  1200   450    50     1

%e   5005 18018 20475  9100  1575    90     1

%e The loop-graphs counted in row n = 3 (loops shown as singletons):

%e   {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}

%e                 {1}{12}{23}  {1}{2}{13}

%e                 {1}{13}{23}  {1}{2}{23}

%e                 {2}{12}{13}  {1}{3}{12}

%e                 {2}{12}{23}  {1}{3}{13}

%e                 {2}{13}{23}  {1}{3}{23}

%e                 {3}{12}{13}  {2}{3}{12}

%e                 {3}{12}{23}  {2}{3}{13}

%e                 {3}{13}{23}  {2}{3}{23}

%t Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]

%t 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 *)

%o (PARI) T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ _Andrew Howroyd_, Jan 14 2024

%Y Row sums are A014068, unlabeled version A000666.

%Y Column k = 0 is A116508, covering version A367863.

%Y The covering case is A368597.

%Y The unlabeled version is A368836.

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

%Y A006125 counts graphs, unlabeled A000088.

%Y A006129 counts covering graphs, unlabeled A002494.

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

%Y A322661 counts labeled covering loop-graphs, connected A062740.

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

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Jan 11 2024