Roller Coaster Permutations and Partition Numbers
William Adamczak
TL;DR
The paper addresses the partitioning complexity of roller coaster permutations $RC_n$ by studying the partition number $P(\pi)$ and defining $P_{max}(n)=\max_{\pi\in RC_n} P(\pi)$. It derives a provable upper bound $P_{max}(n) \le \left\lfloor \frac{\left\lceil \frac{n-2}{2} \right\rceil}{2} \right\rfloor + 2$ by exploiting the strict value separation between even- and odd-indexed elements alongside the recursively alternating structure, and it validates the bound with experimental data for $n<15$ showing near-tightness. The work also conjectures a logarithmic lower bound $P_{max}(n) \ge \left\lfloor \log_2(n) \right\rfloor$, supported by observed doubling patterns and a proposed recurrence $P(2n)=P(n)+1$. These results illuminate that roller coaster permutations, while rich in local extrema, admit relatively small partition coverings and motivate further analysis via ILP and forbidden-subsequence techniques.
Abstract
This paper explores the partition properties of roller coaster permutations, a class of permutations characterized by maximizing the number of alternating runs in all subsequences. We establish a connection between the structure of these permutations and their partition numbers, defined as the minimum number of monotonic subsequences required to cover the permutation. Our main result provides a theoretical upper bound for the partition number of a roller coaster permutation of length $n$, given by $P_{max}(n) \le \lfloor\frac{\lceil\frac{n-2}{2}\rceil}{2}\rfloor + 2$. We further present experimental data for $n < 15$ that suggests this bound is nearly sharp.