Inventory Loops (i.e. Counting Sequences) have Pre-period $2\max S_1+60$

Abstract

An Inventory Sequence $(S_0, S_1, S_2, ...)$ is the iteration of the map $f$ defined roughly by taking an integer to its numericized description (e.g. $f(1381)=211318$ since "$1381$" has two $1$'s, one $3$, and one $8$). Our work analyzes the iteration under the infinite base. Any starting value of positive digits is known to be ultimately periodic [1] (e.g. $S_0=1381$ reaches the 1-cycle $f(3122331418)=3122331418$). Parametrizations of all possible cycles are also known [2,3]. We answer Bronstein and Fraenkel's open question of 26 years showing the pre-period of any such starting value is no more than $2M+60$ where $M=\max S_1$. And oddly the period of the cycle can be determined after only $O(\log\log M)$ iterations.