A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1

Offset

0,6

Comments

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

Formula

E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

Example

Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

Mathematica

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, 2}], {k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n, 0, 5}, {k, 0, Binomial[n+1, 2]}]

Prog

(PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

Crossrefs

Row lengths are A000124.

Diagonal T(n,n-1) is A000272, rooted A000169.

The case without loops is A062734.

Row sums are A062740.

Transpose is A322147.

Column sums are A322151.

Diagonal T(n,n) is A368951, connected case of A368597.

Connected case of A369199, without loops A054548.

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

A000666 counts unlabeled loop-graphs.

A001187 counts connected graphs, unlabeled A001349.

A006125 counts simple graphs, also loop-graphs if shifted left.

A006129 counts covering graphs, unlabeled A002494.

A322661 counts covering loop-graphs, unlabeled A322700.

A368927 counts choosable loop-graphs, covering A369140.

A369141 counts non-choosable loop-graphs, covering A369142.

Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

Sequence in context: A306548 A320531 A345698 * A065719 A336087 A204387

Adjacent sequences: A369192 A369193 A369194 * A369196 A369197 A369198

Keyword

nonn,tabf

Author

Gus Wiseman, Jan 19 2024

Status

approved