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Triangle read by rows: T(n,k) is the number of connected simple graphs G of order n with the property that k is the order of the largest quotient graph G/~ that is a complete graph. 1 <= k <= n.
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%I #19 Apr 02 2024 14:34:53

%S 1,0,1,0,1,1,0,1,4,1,0,1,12,7,1,0,1,23,75,12,1,0,1,24,481,329,17,1

%N Triangle read by rows: T(n,k) is the number of connected simple graphs G of order n with the property that k is the order of the largest quotient graph G/~ that is a complete graph. 1 <= k <= n.

%C Row sums are given by A001349.

%C T(n,2) = 1 is the star graph of order n for n >= 2.

%C T(n,n) = 1 is the complete graph of order n.

%H FindStat, <a href="https://www.findstat.org/StatisticsDatabase/St001670/">St001670: The connected partition number of a graph</a>.

%e n\k | 1 2 3 4 5 6 7

%e ----+------------------------------

%e 1 | 1,

%e 2 | 0, 1

%e 3 | 0, 1, 1

%e 4 | 0, 1, 4, 1

%e 5 | 0, 1, 12, 7, 1

%e 6 | 0, 1, 23, 75, 12, 1

%e 7 | 0, 1, 24, 481, 329, 17, 1

%Y Cf. A001349 (row sums).

%K nonn,tabl,hard,more

%O 1,9

%A _Peter Kagey_, May 06 2023