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

Author

Geoffrey Critzer, Jul 16 2022

Keywords

• nonn
• tabl

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.

Examples

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

Crossrefs

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

Programs

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

Formula

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