login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of topologies on an n-set X such that for all x in X there is a y in X such that x and y are topologically indistinguishable.
3

%I #29 Apr 18 2023 07:36:34

%S 1,0,1,1,10,31,361,2164,32663,313121,6199024,86219497,2225685925,

%T 42396094690,1414152064833,35520966967269,1517860883350266,

%U 48936884016265947,2659543345912283917,107827798819822505332,7409614386025588874195,371626299919138199117981

%N Number of topologies on an n-set X such that for all x in X there is a y in X such that x and y are topologically indistinguishable.

%C Equivalently a(n) is the number of topologies on an n-set X such that for all x in X there is a y in X such that x and y have exactly the same neighborhoods.

%H Pontus von Brömssen, <a href="/A280202/b280202.txt">Table of n, a(n) for n = 0..37</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Topological_indistinguishability">Topological indistinguishability</a>.

%F E.g.f.: A(exp(x) - 1 - x) where A(x) is the e.g.f. for A001035.

%F a(n) = Sum_{k=0..floor(n/2)} A008299(n,k)*A001035(k).

%e a(4) = 10 because letting X = {a,b,c,d} we have the trivial topology; {{},{b,c},{a,d},X} * 3; and {{},{a,b},X} *6.

%Y Column k=0 of A280192.

%Y Cf. A001035, A008299.

%K nonn

%O 0,5

%A _Geoffrey Critzer_, Dec 28 2016

%E a(19)-a(21) from _Pontus von Brömssen_, Apr 05 2023