Larger Digit Permutation (noch nicht übersetzt)

For a positive integer $n$ define $T(n)$ to be the number of strictly larger integers which can be formed by permuting the digits of $n$.

Leading zeros are not allowed and so for $n = 2302$ the total list of permutations would be:

giving $T(2302)=4$.

Further define $S(k)$ to be the sum of $T(n)$ for all $k$-digit numbers $n$. You are given $S(3) = 1701$.

Find $S(12)$.