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.

💡 Research Summary

The paper investigates the notion of arity gap for finite‑valued functions. For an $n$‑ary $k$‑valued function $f$, a variable is called essential if the value of $f$ depends on it. When two distinct essential variables are identified (i.e., forced to take the same value), some of the remaining essential variables may become fictitious (no longer affect the output). The arity gap $gap(f)$ is defined as the smallest number of essential variables that become fictitious under any such identification. While the case $gap(f)=1$ has been extensively studied—especially for Boolean functions using Zhegalkin polynomials—functions with $gap(f)>1$ have remained largely uncharacterised.

The authors first generalise the algebraic representation of $k$‑valued functions. Under the natural restriction $n\le k$, every $n$‑ary $k$‑valued function can be written as a $k$‑ary polynomial in which each monomial is a product of variables and coefficients lie in ${0,\dots,k-1}$. This representation allows the identification operation to be interpreted as a transformation of polynomial coefficients, thereby linking $gap(f)$ to the degree and symmetry properties of the polynomial.

The central contribution is a complete classification of functions satisfying $2\le gap(f)\le n\le k$. The authors prove that any such function must be representable by a symmetric polynomial whose total degree does not exceed $n-r$, where $r=gap(f)$. In other words, the function is a linear combination of elementary symmetric functions of order at most $n-r$. This structural restriction forces the function to belong to one of $\binom{k}{n}$ combinatorial classes determined by the choice of $n$ distinct values from the $k$‑element domain. Within each class, functions are equivalent up to permutation of variables, so the total number of distinct functions can be expressed in closed form.

To count the functions with a given gap $r$, the paper applies the inclusion–exclusion principle to the process of repeatedly identifying variables. Let $N_{k,n}(r)$ denote the number of $n$‑ary $k$‑valued functions with $gap(f)=r$. The authors derive the exact formula \