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A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) 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.

%I A268223 #27 Aug 22 2025 19:58:27
%S A268223 6,21,42,56,168,336,126,504,1512,3024,252,1260,5040,15120,30240,462,
%T A268223 2772,13860,55440,166320,332640,792,5544,33264,166320,665280,1995840,
%U A268223 3991680,1287,10296,72072,432432,2162160,8648640,25945920,51891840
%N A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) 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 A268223 Andrew Howroyd, <a href="/A268223/b268223.txt">Table of n, a(n) for n = 6..1280</a> (first 50 rows)
%H A268223 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 A268223 Triangle begins:
%e A268223     6;
%e A268223    21,   42;
%e A268223    56,  168,  336;
%e A268223   126,  504, 1512,  3024;
%e A268223   252, 1260, 5040, 15120, 30240;
%e A268223   ...
%t A268223 i = 5; Table[ Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
%t A268223 n - i - 1}], {n, 2, 13}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%Y A268223 Row sums give A268219.
%Y A268223 Triangles in this series: A119741, A268217, A268221, A268222, A268223.
%K A268223 nonn,tabl,changed
%O A268223 6,1
%A A268223 _N. J. A. Sloane_, Jan 30 2016
%E A268223 Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017