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

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

0,2

Comments

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.

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A274177 A364375 A075533 * A053814 A293260 A095238

Adjacent sequences: A355780 A355781 A355782 * A355784 A355785 A355786

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jul 16 2022

Status

approved