On finite functions with non-trivial arity gap
  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. 
Slavcho Shtrakov and Joerg Koppitz