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A355783
Triangular array read by rows. T(n,k) is the number of labeled transitive relations on [n] that have exactly k symmetric points.
0
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.
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
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Jul 16 2022
STATUS
approved