A191249 Triangular array T(n,k) read by rows: number of labeled relations of the n-set with exactly k connected components.

2, 12, 4, 432, 72, 8, 61344, 3888, 288, 16, 32866560, 665280, 21600, 960, 32, 68307743232, 407306880, 4328640, 95040, 2880, 64, 561981464819712, 965518299648, 2948037120, 21893760, 362880, 8064, 128

Offset

1,1

Comments

T(n,k) is the number of binary relations R on {1,2,...,n} such that the reflexive, symmetric and transitive closure of R is an equivalence relation with exactly k classes.

Row sums are A002416 = 2^(n^2).

Column 1 is A062738.

T(n,n) = 2^n is the number of binary relations that are a subset of the diagonal relation.

Links

Table of n, a(n) for n=1..28.

Formula

E.g.f. for column k: log(A(x))^k/k! where A(x) is the E.g.f. for A002416

Example

2
12       4
432      72     8
61344    3888   288   16
32866560 665280 21600 960 32

Mathematica

a=Sum[2^(n^2) x^n/n!, {n, 0, 10}];
Transpose[Table[Drop[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], 1], {n, 1, 10}]] // Grid

Crossrefs

Sequence in context: A159323 A038218 A264841 * A333544 A005760 A155892

Adjacent sequences: A191246 A191247 A191248 * A191250 A191251 A191252

Keyword

nonn,tabl

Author

Geoffrey Critzer, May 28 2011

Status

approved