Connecting two types of representations of a permutation of $\F_q$

Abstract: In this paper, we connect two types of representations of a permutation $\sigma$ of the finite field $\F_q$. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and $k$ copies of $x^{q-2}$, for some prescribed value of $k$. The other type is combinatorial, in which the permutation is represented as the composition of a degree-one rational function followed by the product of $k$ $2$-cycles on $\bP^1(\F_q):=\F_q\cup{\infty}$, where each $2$-cycle moves $\infty$. We show that, after modding out by obvious equivalences amongst the algebraic representations, then for each $k$ there is a bijection between the algebraic representations of $\sigma$ and the combinatorial representations of $\sigma$. We also prove analogous results for permutations of $\bP^1(\F_q)$. One consequence is a new characterization of the notion of Carlitz rank of a permutation on $\F_q$, which we use elsewhere to provide an explicit formula for the Carlitz rank. Another consequence involves a classical theorem of Carlitz, which says that if $q>2$ then the group of permutations of $\F_q$ is generated by the permutations induced by degree-one polynomials and $x^{q-2}$. Our bijection provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs.