For integer n (assume n is between 1 and 9), we define lexicographic permutation as a permutation of digits 1, 2, ..., n such that permutation-k (k starting from 0) is greater than the permutation-(k-1).

As an example, for n = 4,

permutation-0 = "1234", permutation-1 = "1243", permutation-2 = "1324", ..., permutation-21 = "4231", permutation-22 = "4312", permutation-23 = "4321".

Define a recursive function that when passed the values of n and k, returns the permutation-k.

You may not use any loops.

Return an empty string if the value of n or k is invalid. Note that n needs to be between 1 and 9, and k will depend on the value of n. We will let you figure out the highest value of k in terms of n.