Codegolf the permanent

Haskell, 59 bytes

a#((b:c):r)=b*p(a++map tail r)+(c:a)#r
_#_=0
p[]=1
p l=[]#l

This does a Laplace-like development along the first column, and uses that the order of the rows doesn't matter. It works for any numeric type.

Input is as list of lists:

Prelude> p [[1,2],[3,4]]
10

J, 5 bytes

+/ .*

J does not offer builtins for the permanent or determinant but instead offers a conjunction u . v y which recursively expands y along the minors and calculates dyadic u . v between the cofactors and the output of the recursive call on the minors. The choices of u and v can vary. For example, using u =: -/ and v =: * is -/ .* which is the determinant. Choices can even by %/ .! where u=: %/, reduce by division, and v =: ! which is binomial coefficient. I'm not sure what that output signifies but you're free to choose your verbs.

An alternative implementation for 47 bytes using the same method in my Mathematica answer.

_1{[:($@]$[:+//.*/)/0,.;@(<@(,0#~<:)"+2^i.@#)"{

This simulates a polynomial with n variables by creating a polynomial with one variable raised to powers of 2. This is held as a coefficient list and polynomial multiplication is performed using convolution, and the index at 2ⁿ will contain the result.

Another implementation for 31 bytes is

+/@({.*1$:\.|:@}.)`(0{,)@.(1=#)

which is a slightly golfed version based on Laplace expansion taken from the J essay on determinants.

Usage

   f =: +/ .*
   f 0 0 $ 0 NB. the empty matrix, create a shape with dimensions 0 x 0
1
   f 0.36697048j0.02459455 0.81148991j0.75269667 0.62568185j0.95950937 , 0.67985923j0.11419187  0.50131790j0.13067928 0.10330161j0.83532727 ,: 0.71085747j0.86199765 0.68902048j0.50886302 0.52729463j0.5974208
_1.7422j2.24768
   f 0.83702504j0.05801749 0.03912260j0.25027115 0.95507961j0.59109069 , 0.07330546j0.8569899 0.47845015j0.45077079 0.80317410j0.5820795 ,: 0.38306447j0.76444045 0.54067092j0.90206306 0.40001631j0.43832931
_1.97212j1.60813
   f 0.61164611j0.42958732 0.69306292j0.94856925 0.4386093j0.04104116 0.92232338j0.32857505 0.40964318j0.59225476 0.69109847j0.32620144 , 0.57851263j0.69458731 0.21746623j0.38778693 0.83334638j0.25805241 0.6485583j0.36137045 0.6589084j0.06557287 0.25411493j0.37812483 , 0.11114704j0.44631335 0.32068031j0.52023283 0.43360984j0.87037973 0.42752697j0.75343656 0.23848512j0.96334466 0.28165516j0.13257001 , 0.66386467j0.21002292 0.11781236j0.00967473 0.75491373j0.44880959 0.66749636j0.90076845 0.0093942j0.06484633 0.21316223j0.4538433 , 0.40175631j0.89340763 0.26849809j0.82500173 0.84124107j0.23030393 0.62689175j0.61870543 0.92430209j0.11914288 0.90655023j0.63096257 ,: 0.85830178j0.16441943 0.91144755j0.49943801 0.5101055j0.60590678 0.51439995j0.37354955 0.79986742j0.87723514 0.43231194j0.54571625
_22.9235j_90.7428

Jelly, 10 9 bytes

Œ!ŒDḢ€P€S

Try it online!

How it works

Œ!ŒDḢ€P€S  Main link. Argument: M (matrix / 2D array)

Œ!         Generate all permutations of M's rows.
  ŒD       Compute the permutations' diagonals, starting with the main diagonal.
    Ḣ€     Head each; extract the main diagonal of each permutation.
      P€   Product each; compute the products of the main diagonals.
        S  Compute the sum of the products.