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A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.

%I A263340 #34 Apr 29 2024 22:59:17
%S A263340 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 A263340 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 A263340 10,4,9,5,2,3,3,2,2,1,0,1,1,0,0,0,1,0,0,0,0,1
%N A263340 Triangle read by rows: T(n,k) is the number of graphs with n vertices containing k triangles.
%C A263340 Row sums give A000088.
%C A263340 First column is A006785.
%C A263340 Row lengths are 1 + binomial(n,3). - _Geoffrey Critzer_, Apr 13 2017
%H A263340 Pontus von Brömssen, <a href="/A263340/b263340.txt">Rows n = 0..10, flattened</a>
%H A263340 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000095">The number of triangles of a graph</a>.
%H A263340 Gus Wiseman, <a href="/A263340/a263340.png">The graphs counted under row n = 5</a>.
%e A263340 Triangle begins:
%e A263340   1;
%e A263340   1;
%e A263340   2;
%e A263340   3,1;
%e A263340   7,2,1,0,1;
%e A263340   14,7,5,2,3,1,0,1,0,0,1;
%e A263340   38,23,28,14,18,9,7,5,4,1,4,1,1,1,0,0,1,0,0,0,1;
%e A263340   ...
%t A263340 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 A263340 Row sums are A000088, labeled A006125.
%Y A263340 Column k = 0 is A006785 (lab A213434), covering A372169 (lab A372168).
%Y A263340 Counting edges gives A008406 (lab A084546), covering A370167 (lab A054548).
%Y A263340 Row lengths are A050407.
%Y A263340 The labeled version is A372170, covering A372167.
%Y A263340 The covering case is A372173, sums A002494, labeled A006129.
%Y A263340 Column k = 1 is A372194 (lab A372172), covering A372174 (lab A372171).
%Y A263340 A001858 counts acyclic graphs, unlabeled A005195.
%Y A263340 A372176 counts labeled graphs by directed cycles, covering A372175.
%Y A263340 Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.
%K A263340 nonn,tabf
%O A263340 0,3
%A A263340 _Christian Stump_, Oct 15 2015
%E A263340 Row 7 from _Geoffrey Critzer_, Apr 13 2017
%E A263340 T(0,0)=1 prepended by _Alois P. Heinz_, Apr 13 2017