How to generate a permutation of n numbers using secret sharing scheme? Consider $n$ parties having their private inputs $s_1,\cdots,s_n$. How can they compute a public function $f(s_1,\cdots,s_n)= \pi$, where $\pi$ is a permutation of numbers between $1,\cdots,n$, using secret sharing scheme?
If you are just lacking a function from some random number to permutations, this is easy:

Consider the set of all permutations of $\{1,\dots,n\}$, put them in lexicographical order and assign natural numbers. There are $n!$ permutations, so let's say $x$ is some random number from $\mathbb{Z}_{n!}$, and here's an algorithm how to get the matching permutation:


Generate list $L = [1,\dots,n]$ and empty list $O = [\;]$
$a = (|L| - 1)!$
$b = \lfloor x / a\rfloor $
Remove $L[b]$ from $L$ and append it to $O$
$x = x \mod a$
if $L$ is not empty: GOTO 2.
output $O$


List indices are starting from $0$ as usual.

It wasn't stated in the question from which set the private inputs were taken, and ideally this would be from $\mathbb{Z}_{n!}$ (or alternatively $\{a, a+1, \dots ,a + n! - 1\}$). Then all you need is to use MPC to calculate the sum $x = s_1 + \dots + s_n \mod n!$, and use above algorithm to calculate the matching permutation. If the entropy of the inputs is not enough, use can use a PRG to expand this input and use $s'_i = PRG(s_i)$ instead (for party $P_i$).