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.
5, 15, 30, 35, 105, 210, 70, 280, 840, 1680, 126, 630, 2520, 7560, 15120, 210, 1260, 6300, 25200, 75600, 151200, 330, 2310, 13860, 69300, 277200, 831600, 1663200, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400, 715, 6435, 51480, 360360, 2162160, 10810800, 43243200, 129729600, 259459200
Offset: 5
Examples
Triangle begins:
5;
15, 30;
35, 105, 210;
70, 280, 840, 1680;
126, 630, 2520, 7560, 15120;
210, 1260, 6300, 25200, 75600, 151200;
...
Links
- Andrew Howroyd, Table of n, a(n) for n = 5..1279 (first 50 rows)
- G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Crossrefs
Programs
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Mathematica
i = 4; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 12}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)
Extensions
Title clarified and more terms added by Geoffrey Critzer, Feb 19 2017
Missing a(19) inserted and a(41) onwards from Andrew Howroyd, Aug 10 2025