Fast arithmetic on matrix elements

Question

I have a square matrix $U$ of dimension $L\times L$, I am interested in the following quantity

$U_{1,j}U_{2,k}U_{3,l}U_{4,m}\cdots U_{L,q}$

summed over all permutations of the letter indices, e.g., $\{j,k,l,m,\cdots q\}$ is an element of Permutations[Range[L]]. What is a fast and efficient way of doing this? I am using the following function:

f[mat_, l_] := Total[Product[Part[mat, j, #[[j]]], {j, l}] & /@ Permutations[Range[l]]]

which is very inefficient. In fact it even runs out of memory for $L>11$

Answer 1

To settle this:

The function being sought by the OP is actually now built-in into Mathematica as Permanent[]. Unfortunately, it is not terribly quick. You can, for instance, use a function like the one in this answer instead.

permanent[mat_?SquareMatrixQ] := Module[{l = Length[mat] - 1, b}, b = 2^l; 
          PadRight[{}, b, {1, -1}].(Times @@@
          ((2 IntegerDigits[2 b - Experimental`GrayCode[l] - 1, 2] - 1).mat))/b]

AbsoluteTiming[permanent[Array[C, {3, 3}]] // Simplify]
   {0.001485, C[1, 3] (C[2, 2] C[3, 1] + C[2, 1] C[3, 2]) +
    C[1, 2] (C[2, 3] C[3, 1] + C[2, 1] C[3, 3]) +
    C[1, 1] (C[2, 3] C[3, 2] + C[2, 2] C[3, 3])}

AbsoluteTiming[Permanent[Array[C, {3, 3}]] // Simplify] (* built-in *)
   {0.005706, C[1, 3] (C[2, 2] C[3, 1] + C[2, 1] C[3, 2]) +
    C[1, 2] (C[2, 3] C[3, 1] + C[2, 1] C[3, 3]) +
    C[1, 1] (C[2, 3] C[3, 2] + C[2, 2] C[3, 3])}