OFFSET
2,1
COMMENTS
A chain topology is a topology that can be totally ordered by inclusion.
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.
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]
EXAMPLE
Triangle begins:
2;
3, 6;
4, 24, 24;
5, 70, 180, 120;
6, 180, 900, 1440, 720;
...
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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Feb 16 2017
STATUS
approved