Counting Finite Maps through Functional Digraphs and Fibers

Abstract

We count solutions of composition identities for maps on finite sets, first for three identities in which both maps vary and then for commutation with a prescribed arithmetic map. For $f\circ g=f\circ g\circ g$, drawing an arrow from each element to its image reduces the enumeration to rooted trees attached to directed cycles. For $f\circ f=f\circ g\circ f$, grouping elements with the same $f$-value leads instead to a partition-based enumeration. A classical rooted-forest enumeration leads in both problems to $\mathcal T(x)$, the exponential generating function for rooted labeled trees. For $f=g\circ f\circ g$, the values on the cycles of $g$ determine all remaining values. The exponential generating function $\mathcal D$ for these ordered pairs and the ordinary generating function $\mathcal A$ for endofunctions up to relabeling satisfy $\mathcal D(x)=\mathcal A(\mathcal T(x))$. Finally, we count functions on $\mathbb Z/n\mathbb Z$ satisfying $f(ax)=af(x)$, first for prime multipliers $a$ and then for a broader class of composite multipliers. In each problem, fixing one map exposes independent choices for the other through its functional digraph or its fibers.