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A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) 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 {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

%I A268217 #43 Aug 22 2025 19:57:08
%S A268217 3,6,12,10,30,60,15,60,180,360,21,105,420,1260,2520,28,168,840,3360,
%T A268217 10080,20160,36,252,1512,7560,30240,90720,181440,45,360,2520,15120,
%U A268217 75600,302400,907200,1814400,55,495,3960,27720,166320,831600,3326400,9979200,19958400
%N A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) 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 {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.
%C A268217 When two leading 0's are added and last element repeated, rows give the coefficients of the path polynomials of the complete graph K_n. - _Eric W. Weisstein_, Jun 04 2017
%H A268217 Andrew Howroyd, <a href="/A268217/b268217.txt">Table of n, a(n) for n = 3..1277</a> (first 50 rows)
%H A268217 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.
%H A268217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.
%H A268217 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphPath.html">Graph Path</a>.
%e A268217 Triangle begins:
%e A268217    3;
%e A268217    6,  12;
%e A268217   10,  30,   60;
%e A268217   15,  60,  180,   360;
%e A268217   21, 105,  420,  1260,   2520;
%e A268217   28, 168,  840,  3360,  10080,  20160;
%e A268217   36, 252, 1512,  7560,  30240,  90720,  181440;
%e A268217   45, 360, 2520, 15120,  75600, 302400,  907200, 1814400;
%e A268217   ...
%t A268217 i = 2; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* _Geoffrey Critzer_, Feb 19 2017 *)
%t A268217 CoefficientList[Table[-(1/2) (n - 1) n x^(n - 2) (Gamma[n - 1] - E^(1/x) Gamma[n - 1, 1/x]), {n, 3, 10}] // FunctionExpand, x] // Flatten (* _Eric W. Weisstein_, Jun 04 2017 *)
%Y A268217 Row sums give A038158.
%Y A268217 Triangles in this series: A119741, A268217, A268221, A268222, A268223.
%Y A268217 Cf. A282507.
%K A268217 nonn,tabl,changed
%O A268217 3,1
%A A268217 _N. J. A. Sloane_, Jan 29 2016
%E A268217 Title clarified by _Geoffrey Critzer_, Feb 19 2017
%E A268217 Corrected and extended by _Andrew Howroyd_, Aug 09 2025