Completely Additive Height Functions: Profile Laws, Matula Bounds, and Inverse Growth

Abstract

The height $H(n)$ of $n$ is the least integer $i$ such that the $i$-th iterate of Euler's totient function $\varphi^{(i)}(n)$ equals $1$. H. N. Shapiro showed that this $H$ is almost completely additive. Building on the fact that this function can be modified to yield a completely additive function, we establish a general correspondence: to every multi-partition structure there corresponds a completely additive function. In this paper, a \emph{height function} is a completely additive map $H:\mathbb{N}\to\mathbb{N}_0$ with $H(1)=0$ whose prime fibres $\{p:\,H(p)=k\}$ are finite for every $k\ge1$. Writing \[ \pi_k=\#\{p:\,H(p)=k\},\qquad N_k=\#\{n:\,H(n)=k\}, \] complete additivity forces the identity \[ \sum_{k\ge0}N_k q^k \;=\; \prod_{j\ge1}(1-q^j)^{-\pi_j}. \] Thus, the prime--height profile $(\pi_k)$ canonically determines the height multiplicities $(N_k)$, linking to the asymptotic theory of weighted partitions. We introduce a broad class of iteratively defined heights on primes, encompassing Matula-type heights (encoding rooted trees) and Shapiro-type totient heights, and show they extend to genuine height functions. In the Matula case this yields a purely number-theoretic proof of the classical extremal bounds for minimal and maximal Matula numbers, answering a question of Gutman and Ivi\'c without recourse to graph theory. Using Meinardus' theorem we prove an \emph{inverse-growth} principle in the polynomial regime: if $\Pi(x)=\sum_{j\le x}\pi_j \sim (C/\alpha)x^\alpha$, then $\log N_k$ satisfies a stretched-exponential law with an explicit constant, and conversely under a standard Tauberian hypothesis. We further derive average-order consequences in this regime for a canonical sequential realization of a given profile. Finally, we briefly discuss behavior beyond the polynomial setting, with computations in the Shapiro case suggesting substantially richer phenomena.