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A280192 Triangle read by rows: T(n,k) = number of topologies on an n-set X such that there are exactly k elements in X that are topologically distinguishable, n >= 0, 0 <= k <= n.

%I A280192 #41 Apr 20 2023 11:26:38
%S A280192 1,0,1,1,0,3,1,9,0,19,10,12,114,0,219,31,300,190,2190,0,4231,361,1158,
%T A280192 10140,4380,63465,0,130023,2164,26341,46389,451920,148085,2730483,0,
%U A280192 6129859,32663,192496,1930852,2381624,27601000,7281288,171636052,0,431723379
%N A280192 Triangle read by rows: T(n,k) = number of topologies on an n-set X such that there are exactly k elements in X that are topologically distinguishable, n >= 0, 0 <= k <= n.
%C A280192 T(n,0) = A280202(n) is the number of topologies on an n-set X such that for all x in X there exists a y in X such that x and y have exactly the same neighborhoods.
%C A280192 Equivalently, T(n,k) is the number of labeled quasi-orders R on [n] with exactly k singletons in the equivalence relation R intersect R^(-1), cf. Schein link. - _Geoffrey Critzer_, Apr 18 2023
%H A280192 Alois P. Heinz, <a href="/A280192/b280192.txt">Rows n = 0..18, flattened</a>
%H A280192 B. M. Schein, <a href="https://doi.org/10.3792/pja/1195520400">A construction for idempotent binary relations</a>, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.
%H A280192 Wikipedia, <a href="https://en.wikipedia.org/wiki/Topological_indistinguishability">Topological indistinguishability</a>.
%F A280192 E.g.f.: A(exp(x) - 1 - x + y*x) where A(x) is the e.g.f. for A001035.
%F A280192 Sum_{k=0..n} T(n,k)*2^k = A006905(n). - _Geoffrey Critzer_, Apr 18 2023
%e A280192 Triangle begins:
%e A280192      1;
%e A280192      0,     1;
%e A280192      1,     0,     3;
%e A280192      1,     9,     0,     19;
%e A280192     10,    12,   114,      0,    219;
%e A280192     31,   300,   190,   2190,      0,    4231;
%e A280192    361,  1158, 10140,   4380,  63465,       0, 130023;
%e A280192   2164, 26341, 46389, 451920, 148085, 2730483,      0, 6129859;
%e A280192   ...
%t A280192 A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]];
%t A280192 lg = Length[A001035];
%t A280192 A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
%t A280192 CoefficientList[#, y]& /@ (CoefficientList[A[Exp[x] - 1 - x + y*x] + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* _Jean-François Alcover_, Jan 01 2020 *)
%Y A280192 Right border gives A001035.
%Y A280192 Row sums give A000798.
%Y A280192 Column k=0 gives A280202.
%Y A280192 Cf. A006905.
%K A280192 nonn,tabl
%O A280192 0,6
%A A280192 _Geoffrey Critzer_, Dec 28 2016