  Given an $n$-ary
  $k-$valued function $f$, $gap(f)$ denotes the minimal number of essential
variables in $f$ which become fictive when identifying any two distinct
essential variables in $f$.
  We particularly solve a problem concerning the explicit determination of
$n$-ary
  $k-$valued functions $f$ with $2\leq gap(f)\leq n\leq k$. Our methods yield
new combinatorial results about the number of such functions.
