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:

$2023,2032,2203,2230,\mathbf{2302},2320,3022,3202,3220$

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 $S(12)$.