cp's OEIS Frontend

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

%I A191249 #10 Jan 17 2013 12:39:27
%S A191249 2,12,4,432,72,8,61344,3888,288,16,32866560,665280,21600,960,32,
%T A191249 68307743232,407306880,4328640,95040,2880,64,561981464819712,
%U A191249 965518299648,2948037120,21893760,362880,8064,128
%N A191249 Triangular array T(n,k) read by rows:  number of labeled relations of the n-set with exactly k connected components.
%C A191249 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 A191249 Row sums are A002416 = 2^(n^2).
%C A191249 Column 1 is A062738.
%C A191249 T(n,n) = 2^n is the number of binary relations that are a subset of the diagonal relation.
%F A191249 E.g.f. for column k: log(A(x))^k/k! where A(x) is the E.g.f. for A002416
%e A191249 2
%e A191249 12       4
%e A191249 432      72     8
%e A191249 61344    3888   288   16
%e A191249 32866560 665280 21600 960 32
%t A191249 a=Sum[2^(n^2) x^n/n!,{n,0,10}];
%t A191249 Transpose[Table[Drop[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}],x],1], {n, 1, 10}]] // Grid
%K A191249 nonn,tabl
%O A191249 1,1
%A A191249 _Geoffrey Critzer_, May 28 2011