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A084268 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

%I A084268 #56 Feb 16 2025 08:32:49
%S A084268 1,1,1,1,2,1,1,6,3,1,1,12,16,4,1,1,34,84,31,5,1,1,87,579,318,52,6,1,1,
%T A084268 302,5721,5366,867,81,7,1,1,1118,87381,155291,28722,2028,118,8,1,1,
%U A084268 5478,2104349,7855628,1919895,115391,4251,165,9,1,1,32302,78315231,675054876,250530482,14662562,393963,8214,222,10,1
%N A084268 Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.
%C A084268 T(n,1) = T(n,n) = 1 (here we count the empty graph and the complete graph).  T(n,n-1) = n-1 (here we count the graphs with clique number equal to n-1). - _Geoffrey Critzer_, Oct 12 2016
%C A084268 Row sums give A000088. - _Joerg Arndt_, Oct 13 2016
%H A084268 Keith Briggs, <a href="http://keithbriggs.info/cgt.html">combinatorial graph theory</a>, see entry "number of graphs on n nodes with clique number k".
%H A084268 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000098">The chromatic number of a graph.</a>
%H A084268 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>
%H A084268 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/n-ChromaticGraph.html">n-Chromatic Graph</a>
%e A084268 Triangle begins:
%e A084268   1;
%e A084268   1,    1;
%e A084268   1,    2,       1;
%e A084268   1,    6,       3,       1;
%e A084268   1,   12,      16,       4,       1;
%e A084268   1,   34,      84,      31,       5,      1;
%e A084268   1,   87,     579,     318,      52,      6,    1;
%e A084268   1,  302,    5721,    5366,     867,     81,    7,   1;
%e A084268   1, 1118,   87381,  155291,   28722,   2028,  118,   8, 1;
%e A084268   1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1;
%e A084268   ...
%o A084268 (Sage) # prints triangle with a leading zero in each row
%o A084268 for n in range(1, 8) :
%o A084268     st = [0 for j in range(n+1)]
%o A084268     G = graphs(n)
%o A084268     for g in G :
%o A084268         st[ g.chromatic_number() ] += 1
%o A084268     print(st)
%o A084268 # _Joerg Arndt_, Oct 13 2016
%Y A084268 Columns k=1..9 are A057427, A076278, A076279, A076280, A076281, A076282, A076283, A205567, A205568.
%Y A084268 Partial row sums include A033995, A076315, A076316, A076317, A076318, A076319, A076320, A076321.
%Y A084268 Row sums are A000088.
%Y A084268 Cf. A084269 (connected), A115597 (essentially the same sequence).
%K A084268 nonn,tabl
%O A084268 1,5
%A A084268 _Eric W. Weisstein_, May 24 2003
%E A084268 Offset corrected by _Joerg Arndt_, Oct 13 2016
%E A084268 a(36)-a(55) from _Joerg Arndt_, Oct 15 2016
%E A084268 a(56)-a(66) from _Andrew Howroyd_, Dec 02 2018