A263340 - OEIS

%I #34 Apr 29 2024 22:59:17

%S 1,1,2,3,1,7,2,1,0,1,14,7,5,2,3,1,0,1,0,0,1,38,23,28,14,18,9,7,5,4,1,

%T 4,1,1,1,0,0,1,0,0,0,1,107,102,141,117,123,92,80,63,49,35,35,23,15,17,

%U 10,4,9,5,2,3,3,2,2,1,0,1,1,0,0,0,1,0,0,0,0,1

%N Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

%C Row sums give A000088.

%C First column is A006785.

%C Row lengths are 1 + binomial(n,3). - _Geoffrey Critzer_, Apr 13 2017

%H Pontus von Brömssen, <a href="/A263340/b263340.txt">Rows n = 0..10, flattened</a>

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000095">The number of triangles of a graph</a>.

%H Gus Wiseman, <a href="/A263340/a263340.png">The graphs counted under row n = 5</a>.

%e Triangle begins:

%e   1;

%e   1;

%e   2;

%e   3,1;

%e   7,2,1,0,1;

%e   14,7,5,2,3,1,0,1,0,0,1;

%e   38,23,28,14,18,9,7,5,4,1,4,1,1,1,0,0,1,0,0,0,1;

%e   ...

%t Table[Table[Count[Table[Tr[MatrixPower[AdjacencyMatrix[GraphData[{n, i}]], 3]]/6, {i, 1, NumberOfGraphs[n]}], k], {k, 0, Binomial[n, 3]}], {n, 1, 7}] (* _Geoffrey Critzer_, Apr 13 2017 *)

%Y Row sums are A000088, labeled A006125.

%Y Column k = 0 is A006785 (lab A213434), covering A372169 (lab A372168).

%Y Counting edges gives A008406 (lab A084546), covering A370167 (lab A054548).

%Y Row lengths are A050407.

%Y The labeled version is A372170, covering A372167.

%Y The covering case is A372173, sums A002494, labeled A006129.

%Y Column k = 1 is A372194 (lab A372172), covering A372174 (lab A372171).

%Y A001858 counts acyclic graphs, unlabeled A005195.

%Y A372176 counts labeled graphs by directed cycles, covering A372175.

%Y Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.

%K nonn,tabf

%O 0,3

%A _Christian Stump_, Oct 15 2015

%E Row 7 from _Geoffrey Critzer_, Apr 13 2017

%E T(0,0)=1 prepended by _Alois P. Heinz_, Apr 13 2017