Smooth permutations and polynomials revisited
Ofir Gorodetsky
TL;DR
This work derives sharp, uniform asymptotics for the probability that a random permutation on $n$ elements is $m$-smooth, tying the main term to a shifted Dickman function $\rho(\cdot)$ and achieving error terms parallel to de Bruijn’s classical integers setting. Extending to polynomials over finite fields, the authors identify a phase transition near $m \approx (3/2)\log_q n$ in the $m$-smooth regime, quantified via generating functions $F_q$, $F$, and $G_q$ and a transition factor $\exp(-A)$. Across both settings, a detailed saddle-point framework coupled with careful control of $T(s)$ and related functions yields precise expansions and explicit error bounds, including a refinement to replace the main term by $\rho(n/(m+1/2))$ where appropriate. The results unify the integer- and function-field cases, extend known ranges, and expose regimes where the smooth-probability transitions dramatically, with implications for counting and randomness in combinatorial structures.
Abstract
We study the counts of smooth permutations and smooth polynomials over finite fields. For both counts we prove an estimate with an error term that matches the error term found in the integer setting by de Bruijn more than 70 years ago. The main term is the usual Dickman $ρ$ function, but with its argument shifted. We determine the order of magnitude of $\log(p_{n,m}/ρ(n/m))$ where $p_{n,m}$ is the probability that a permutation on $n$ elements, chosen uniformly at random, is $m$-smooth. We uncover a phase transition in the polynomial setting: the probability that a polynomial of degree $n$ in $\mathbb{F}_q$ is $m$-smooth changes its behavior at $m\approx (3/2)\log_q n$.