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A268221 Triangle read by rows: T(n,k) (n>=4, 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 {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.

%I A268221 #28 Aug 22 2025 19:57:28
%S A268221 4,10,20,20,60,120,35,140,420,840,56,280,1120,3360,6720,84,504,2520,
%T A268221 10080,30240,60480,120,840,5040,25200,100800,302400,604800,165,1320,
%U A268221 9240,55440,277200,1108800,3326400,6652800,220,1980,15840,110880,665280,3326400,13305600,39916800,79833600
%N A268221 Triangle read by rows: T(n,k) (n>=4, 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 {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.
%H A268221 Andrew Howroyd, <a href="/A268221/b268221.txt">Table of n, a(n) for n = 4..1278</a> (first 50 rows)
%H A268221 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 A268221 Triangle begins:
%e A268221     4;
%e A268221    10,  20;
%e A268221    20,  60,  120;
%e A268221    35, 140,  420,   840;
%e A268221    56, 280, 1120,  3360,   6720;
%e A268221    84, 504, 2520, 10080,  30240,  60480;
%e A268221   120, 840, 5040, 25200, 100800, 302400, 604800;
%e A268221   ...
%t A268221 i = 3; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 12}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%Y A268221 Row sums give A268218.
%Y A268221 Triangles in this series: A119741, A268217, A268221, A268222, A268223.
%Y A268221 Cf. A282507.
%K A268221 nonn,tabl,changed
%O A268221 4,1
%A A268221 _N. J. A. Sloane_, Jan 30 2016
%E A268221 Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017