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.
2, 3, 6, 4, 24, 24, 5, 70, 180, 120, 6, 180, 900, 1440, 720, 7, 434, 3780, 10920, 12600, 5040, 8, 1008, 14448, 67200, 134400, 120960, 40320, 9, 2286, 52164, 367416, 1134000, 1723680, 1270080, 362880, 10, 5100, 181500, 1864800, 8341200, 19051200, 23284800, 14515200, 3628800
Offset: 2
Examples
Triangle begins: 2; 3, 6; 4, 24, 24; 5, 70, 180, 120; 6, 180, 900, 1440, 720; ...
Links
- Michael De Vlieger, Table of n, a(n) for n = 2..1226 (rows 2..50, flattened)
- Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.
Programs
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Mathematica
nn = 10; Map[Select[#, # > 0 &] &, Drop[Range[0, nn]! CoefficientList[Series[x/(1 - y (Exp[x] - 1)), {x, 0, nn}], {x, y}], 2]] // Grid
Formula
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)).
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]
Comments