On finite functions with non-trivial arity gap

TL;DR

This paper investigates the properties of finite functions with a focus on the arity gap, providing explicit characterizations and combinatorial counts for functions with specific gap values.

Contribution

It offers a complete solution to determining n-ary k-valued functions with a given arity gap and introduces new combinatorial results related to these functions.

Findings

Explicit formulas for functions with 2 ≤ gap(f) ≤ n ≤ k

New combinatorial counts for such functions

Characterization of functions based on their arity gap

Abstract

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.