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A282507 Triangular array read by rows. T(n,k) is the number of chain topologies on an n-set with exactly k open sets where one of the open sets is a single point set, n >= 2, 3 <= k <= n+1.

%I A282507 #30 Aug 22 2025 19:58:48
%S A282507 2,3,6,4,24,24,5,70,180,120,6,180,900,1440,720,7,434,3780,10920,12600,
%T A282507 5040,8,1008,14448,67200,134400,120960,40320,9,2286,52164,367416,
%U A282507 1134000,1723680,1270080,362880,10,5100,181500,1864800,8341200,19051200,23284800,14515200,3628800
%N A282507 Triangular array read by rows. T(n,k) is the number of chain topologies on an n-set with exactly k open sets where one of the open sets is a single point set, n >= 2, 3 <= k <= n+1.
%C A282507 A chain topology is a topology that can be totally ordered by inclusion.
%H A282507 Michael De Vlieger, <a href="/A282507/b282507.txt">Table of n, a(n) for n = 2..1226</a> (rows 2..50, flattened)
%H A282507 Loïc Foissy, <a href="https://arxiv.org/abs/2304.00810">Hopf algebraic structures on hypergraphs and multi-complexes</a>, arXiv:2304.00810 [math.CO], 2023.
%F A282507 E.g.f.: y^2*x/(1 - y*(exp(x) - 1)). Generally for chain topologies where the smallest nonempty open set has size m: (x^m/m!) * y^2/(1 - y*(exp(x) - 1)).
%F A282507 A conjecture I made to Loic Foissy, who replied: sequence T(n,k) counts surjective maps [n]->> [k] such that k is obtained exactly once, whereas sequence A019538 b(n,k) counts surjective maps [n]->> [k]. To construct an element for T(n,k), you may choose the element of [n] giving k (n choices), then a surjection from the n-1 remaining elements to [k-1] (b(n-1,k-1) choices). This gives T(n,k) = n * b(n-1,k-1), if k,n>1. - _Tom Copeland_, Nov 10 2023 [So it is now a theorem, not a conjecture, right? - _N. J. A. Sloane_, Dec 23 2023]
%e A282507 Triangle begins:
%e A282507   2;
%e A282507   3,   6;
%e A282507   4,  24,  24;
%e A282507   5,  70, 180,  120;
%e A282507   6, 180, 900, 1440, 720;
%e A282507   ...
%t A282507 nn = 10; Map[Select[#, # > 0 &] &, Drop[Range[0, nn]! CoefficientList[Series[x/(1 - y (Exp[x] - 1)), {x, 0, nn}], {x, y}], 2]] // Grid
%Y A282507 Cf. A119741 where the topologies are further restricted.
%Y A282507 Row sums = A052882.
%Y A282507 Cf. A019538.
%K A282507 nonn,tabl,changed
%O A282507 2,1
%A A282507 _Geoffrey Critzer_, Feb 16 2017