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Triangle read by rows: T(n,k) (n>=2, 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 {0,1,2,...,s,n} where s is the size of the largest proper open set in t.
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%I #44 Jan 28 2022 07:43:22

%S 2,3,6,4,12,24,5,20,60,120,6,30,120,360,720,7,42,240,840,2520,5040,8,

%T 56,336,1680,6720,20160,40320,9,72,504,3024,15120,60480,181440,362880,

%U 10,90,720,5040,30240,151200,604800,1814400,3628800

%N Triangle read by rows: T(n,k) (n>=2, 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 {0,1,2,...,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 2;

%e 3, 6;

%e 4, 12, 24;

%e 5, 20, 60, 120;

%e 6, 30, 120, 360, 720;

%e 7, 42, 240, 840, 2520, 5040;

%e 8, 56, 336, 1680, 6720, 20160, 40320;

%e 9, 72, 504, 3024, 15120, 60480, 181440, 362880;

%e 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800;

%e ...

%t i = 1; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)

%Y Row sums give A038156. A008279 (main entry).

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

%Y Cf. A282507.

%K nonn,tabl

%O 2,1

%A _N. J. A. Sloane_, Jan 29 2016

%E Definition clarified by _Geoffrey Critzer_, Feb 19 2017