A268222 - OEIS

%I #21 Jan 08 2018 01:58:32

%S 5,15,30,35,105,210,70,280,840,1680,126,630,2520,7560,210,1260,6300,

%T 25200,75600,151200,330,2310,13860,69300,277200,831600,1663200,495,

%U 3960,27720,166320,831600,3326400,9979200,19958400

%N Triangle read by rows: T(n,k) (n>=5, 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 {0,4,5,6,...,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&amp;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 5,

%e 15,30,

%e 35,105,210,

%e 70,280,840,1680,

%e 126,630,2520,7560,15120,

%e 210,1260,6300,25200,75600,151200,

%e ...

%t i = 4; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,

%t n - i - 1}], {n, 2, 12}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)

%Y Row sums give A268219.

%Y Triangles in this series: A268216, A268217, A268221, A268222, A268223.

%Y Cf. A282507.

%K nonn,tabl,more

%O 5,1

%A _N. J. A. Sloane_, Jan 30 2016

%E Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017