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Triangular array T(n,k) read by rows: number of labeled relations of the n-set with exactly k connected components.
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%I #10 Jan 17 2013 12:39:27

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

%T 68307743232,407306880,4328640,95040,2880,64,561981464819712,

%U 965518299648,2948037120,21893760,362880,8064,128

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

%C 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.

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

%C Column 1 is A062738.

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

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

%e 2

%e 12 4

%e 432 72 8

%e 61344 3888 288 16

%e 32866560 665280 21600 960 32

%t a=Sum[2^(n^2) x^n/n!,{n,0,10}];

%t Transpose[Table[Drop[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}],x],1], {n, 1, 10}]] // Grid

%K nonn,tabl

%O 1,1

%A _Geoffrey Critzer_, May 28 2011