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.

1, 0, 1, 1, 0, 3, 1, 9, 0, 19, 10, 12, 114, 0, 219, 31, 300, 190, 2190, 0, 4231, 361, 1158, 10140, 4380, 63465, 0, 130023, 2164, 26341, 46389, 451920, 148085, 2730483, 0, 6129859, 32663, 192496, 1930852, 2381624, 27601000, 7281288, 171636052, 0, 431723379

Offset: 0

Author

Geoffrey Critzer, Dec 28 2016

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			Triangle begins:
     1;
     0,     1;
     1,     0,     3;
     1,     9,     0,     19;
    10,    12,   114,      0,    219;
    31,   300,   190,   2190,      0,    4231;
   361,  1158, 10140,   4380,  63465,       0, 130023;
  2164, 26341, 46389, 451920, 148085, 2730483,      0, 6129859;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..18, flattened
• B. M. Schein, A construction for idempotent binary relations, Proc. Japan Acad., Vol. 46, No. 3 (1970), pp. 246-247.
• Wikipedia, Topological indistinguishability.

Crossrefs

Right border gives A001035.

Row sums give A000798.

Column k=0 gives A280202.

Cf. A006905.

Programs

Mathematica

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
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 *)

Formula

E.g.f.: A(exp(x) - 1 - x + y*x) where A(x) is the e.g.f. for A001035.

Sum_{k=0..n} T(n,k)*2^k = A006905(n). - Geoffrey Critzer, Apr 18 2023