arXiv:2611.08674v1 Announce Type: new 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.
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