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A268222 Triangle read by rows: T(n,k) (n>=5, k=3..n-2) 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.

%I A268222 #29 Aug 22 2025 19:57:52
%S A268222 5,15,30,35,105,210,70,280,840,1680,126,630,2520,7560,15120,210,1260,
%T A268222 6300,25200,75600,151200,330,2310,13860,69300,277200,831600,1663200,
%U A268222 495,3960,27720,166320,831600,3326400,9979200,19958400,715,6435,51480,360360,2162160,10810800,43243200,129729600,259459200
%N A268222 Triangle read by rows: T(n,k) (n>=5, k=3..n-2) 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 A268222 Andrew Howroyd, <a href="/A268222/b268222.txt">Table of n, a(n) for n = 5..1279</a> (first 50 rows)
%H A268222 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 A268222 Triangle begins:
%e A268222     5;
%e A268222    15,   30;
%e A268222    35,  105,  210;
%e A268222    70,  280,  840,  1680;
%e A268222   126,  630, 2520,  7560, 15120;
%e A268222   210, 1260, 6300, 25200, 75600, 151200;
%e A268222 ...
%t A268222 i = 4; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
%t A268222 n - i - 1}], {n, 2, 12}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%Y A268222 Row sums give A268219.
%Y A268222 Triangles in this series: A119741, A268217, A268221, A268222, A268223.
%Y A268222 Cf. A282507.
%K A268222 nonn,tabl,changed
%O A268222 5,1
%A A268222 _N. J. A. Sloane_, Jan 30 2016
%E A268222 Title clarified and more terms added by _Geoffrey Critzer_, Feb 19 2017
%E A268222 Missing a(19) inserted and a(41) onwards from _Andrew Howroyd_, Aug 10 2025