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%I #19 Jan 08 2018 01:58:33
%S 6,21,42,56,168,336,126,504,1512,3024,252,1260,5040,15120,30240,462,
%T 2772,13860,55440,166320,332640,792,5544,33264,166320,665280,1995840,
%U 3991680,1287,10296,72072,432432,2162160,8648640,25945920,51891840
%N Triangle read by rows: T(n,k) (n>=6, k=3..n+1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.
%H G. A. Kamel, <a href="http://www.aascit.org/journal/archive2?journalId=928&paperId=2310">Partial Chain Topologies on Finite Sets</a>, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
%e Triangle begins:
%e 6,
%e 21,42,
%e 56,168,336,
%e 126,504,1512,3024,
%e 252,1260,5040,15120,30240,
%e ...
%t i = 5; Table[ Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
%t n - i - 1}], {n, 2, 13}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%Y Row sums give A268219.
%Y Triangles in this series: A268216, A268217, A268221, A268222, A268223.
%K nonn,tabl,more
%O 6,1
%A _N. J. A. Sloane_, Jan 30 2016
%E Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017
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