The Long and the Short on Counting Sequences

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

“…They asked a few follow-up questions. One about a different variation was answered by Sauerberg and Shu in 1997 [3]. The only question about our present variation was: how long will a given starting value take to reach an Inventory Loop?…”

“…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 .…”

“…-all the aforementioned variations have been studied. The first question is really asking whether our base is infinite (see [1,2,3]) or finite (see [5,6]). There are also perfectly reasonable infinite starting values [1].…”

“…There are also perfectly reasonable infinite starting values [1]. The "number" 2 2132231455565758696 (10)... has two 1's, three 2's, two 3's, one 4, five 5's, five 6's, five 7's, five 8's, six 9's, six 10's, ... Digits can be counted smallest-to-largest (see [1,2,3,8,9]), by appearance (see [5,6,7]), or largest-to-smallest (which isn't much different from smallest-to-largest). And we have been counting all digits at once (see [1,2,3,6,8,9]) but the most famous variation -Conway's look-and-say sequence [4,7] -counts only consecutive runs.…”

“…That is, the name→variation map is ill-defined in both directions. Our variation of present concern is studied in [1,2,3]:…”