Inverse permutation index

Mathematica, 74 bytes

Max@k[i,Flatten@Outer[i=Permutations[j=Range@#];k=Position,{i[[#]]},j,1]]&

Uses 1-indexing. Very inefficient. (uses ~11GB of memory when the input is 11)

Explanation

j=Range@#

Generate a list from 1 to N. Store that in j.

i=Permutations[...]

Find all permutations of j. Store that in i.

k=Position

Store the Position function in k. (to reduce byte-count when using Position again)

Flatten@Outer[...,{i[[#]]},j,1]

Find the inverse permutation of the N-th permutation.

Max@k[i,...]

Find the k (Position) of the inverse permutation in i (all permutations)

Using built-ins, 46 43 bytes

a[(a=Ordering)/@Permutations@Range@#][[#]]&

1-indexed.

05AB1E, 14 13 bytes

Very memory inefficient. Now even more memory inefficient (but 1 byte shorter).
0-based range.
Uses CP-1252 encoding.

ƒ¹ÝœD¹èNkˆ}¯k

Try it online! or as a Modified test suite

Explanation

ƒ               # for N in range[0 .. x]
 ¹ÝœD           # generate 2 copies of all permutations of range[0 .. x]
     ¹è         # get permutation at index x
       Nkˆ      # store index of N in that permutation in global list
         }      # end loop
          ¯k    # get index of global list (inverse) in list of permutations

Jelly, 6 bytes

ịŒ!⁺iR

I/O uses 1-based indexing. Very slow and memory-hungry.

Verification

As long as the input does not exceed 8! = 40320, it is sufficient to consider all permutations of the array [1, …, 8]. For the last test case, the permutations of [1, …, 9] suffice.

With slightly modified code that only considers the permutations of the first 8 or 9 positive integers, you can Try it online! or verify all remaining test cases.

How it works

ịŒ!⁺iR  Main link. Argument: n

 Œ!     Yield all permutations of [1, ..., n].
ị       At-index; retrieve the n-th permutation.
   ⁺    Duplicate the Œ! atom, generating all permutations of the n-th permutation.
     R  Range; yield [1, ..., n].
    i   Index; find the index of [1, ..., n] in the generated 2D array.

Alternate approach, 6 bytes (invalid)

Œ!Ụ€Ụi

It's just as long and it uses the forbidden grade up atom Ụ, but it's (arguably) more idiomatic.

By prepending 8 (or 9 for the last test case), we can actually Try it online!

How it works

Œ!Ụ€Ụi  Main link. Argument: n

Œ!      Yield all permutations of [1, ..., n].
  Ụ€    Grade up each; sort the indices of each permutation by the corresponding
        values. For a permutation of [1, ..., n], this inverts the permutation.
    Ụ   Grade up; sort [1, ..., n!] by the corresponding inverted permutations
        (lexicographical order).
     i  Index; yield the 1-based index of n, which corresponds to the inverse of
        the n-th permutation.