login
Triangular array read by rows: T(n,k) is the number of elements x in {1,2,...,n} such that |(f^-1)(x)| = k over all functions f:{1,2,...,n}->{1,2,...,n}; n>=0, 0<=k<=n.
1

%I #16 Jan 22 2013 19:28:32

%S 0,0,1,2,4,2,24,36,18,3,324,432,216,48,4,5120,6400,3200,800,100,5,

%T 93750,112500,56250,15000,2250,180,6,1959552,2286144,1143072,317520,

%U 52920,5292,294,7,46118408,52706752,26353376,7529536,1344560,153664,10976,448,8

%N Triangular array read by rows: T(n,k) is the number of elements x in {1,2,...,n} such that |(f^-1)(x)| = k over all functions f:{1,2,...,n}->{1,2,...,n}; n>=0, 0<=k<=n.

%C Row sums = n^(n+1) = Sum_{k=1..n} T(n,k)*k.

%C Column for k=0 is A209290.

%C Distribution is Poisson with mean = 1.

%H Alois P. Heinz, <a href="/A210457/b210457.txt">Rows n = 0..140, flattened</a>

%F T(n,k) = binomial(n,k)*(n-1)^(n-k)*n.

%e 0;

%e 0, 1;

%e 2, 4, 2;

%e 24, 36, 18, 3;

%e 324, 432, 216, 48, 4;

%e 5120, 6400, 3200, 800, 100, 5;

%e 93750, 112500, 56250, 15000, 2250, 180, 6;

%e 1959552, 2286144, 1143072, 317520, 52920, 5292, 294, 7;

%t nn=7;f[list_]:=Sum[list[[i]]*(i-1),{i,1,Length[list]}];g[list_]:= Select[list,#>0&];Prepend[Insert[Map[g,Transpose[Table[Table[f[ CoefficientList[n!Coefficient[Series[(Exp[x]-x^m/m!+y x^m/m!)^n,{x,0,nn}],x^n],y]],{n,1,nn}],{m,0,nn}]]],0,{1,1}],{0}]//Grid

%K nonn,tabl

%O 0,4

%A _Geoffrey Critzer_, Jan 21 2013