On the Number of Operations in a Clone

A clone C on a set A is a set of operations on A containing the projection operations and closed under composition. A combinatorial invariant of a clone is its p(n)-sequence ,where p(n)(C) is the number of essentially n-ary operations in C. We investigate the links between this invariant and structural properties of clones. It has been conjectured that the p(n)-sequence of a clone on a finite set is either eventually strictly increasing or is bounded above by a finite constant. We verify this conjecture for a large family of clones. A special role in our work is played by totally symmetric operations and totally symmetric clones. We show that every totally symmetric clone on a finite set has a bounded p(n)-sequence and that it is decidable if a clone is totally symmetric.