A280202 -id:A280202 - cp's OEIS frontend

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

Geoffrey Critzer, Dec 28 2016

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

			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;
  ...

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.

Right border gives A001035.
Row sums give A000798.
Column k=0 gives A280202.
Cf. A006905.

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 *)

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

A355783 Triangular array read by rows. T(n,k) is the number of labeled transitive relations on [n] that have exactly k symmetric points.

Original entry on oeis.org

1, 2, 0, 12, 0, 1, 152, 0, 18, 1, 3504, 0, 456, 24, 10, 135392, 0, 17520, 760, 600, 31, 8321472, 0, 1015440, 35040, 40560, 2316, 361, 784621952, 0, 87375456, 2369360, 3615360, 185556, 52682, 2164, 110521185024, 0, 10984707328, 233001216, 441616000, 19052992, 7723408, 384992, 32663

Offset: 0

Geoffrey Critzer, Jul 16 2022

Let R be a binary relation on [n]. Then x in [n] is a symmetric point of R if there is a y in [n] with x != y and both (x,y),(y,x) in R.

			       1,
       2, 0,
      12, 0,     1,
     152, 0,    18,   1,
    3504, 0,   456,  24,  10,
  135392, 0, 17520, 760, 600, 31

Cf. A280202 (main diagonal), A085628 (column k=0), A006905 (row sums).

nn = 18; A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt",
    "Table"], {, }][[All, 2]]; A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, nn}];
Table[Take[(Range[0, nn]! CoefficientList[Series[A[Exp[y x] - 1 - y x + x + x], {x, 0, nn}], {x,y}])[[i]], i], {i, 1, nn}] // Grid

E.g.f.: A(exp(y*x) - 1 - y*x + 2*x) where A(x) is the e.g.f. for A001035.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula