A new upper bound to (a variant of) the pancake problem

TL;DR

This paper establishes a new upper bound for the pancake problem variant involving prefix and suffix reversals, showing that sorting permutations can be achieved within 1.5 times the permutation size, highlighting a natural barrier for future improvements.

Contribution

The paper introduces a human-proof upper bound of 1.5k + O(1) for the pancake problem with prefix and suffix reversals, indicating a fundamental limit for this problem variant.

Findings

The lower bound for h(k) remains at (15/14)k - O(1).

The upper bound for h(k) is proven to be 1.5k + O(1).

The constant 1.5 is a natural barrier for this problem variant.

Abstract

The "pancake problem" asks how many prefix reversals are sufficient to sort any permutation $\pi \in \mathcal{S}_k$ to the identity. We write $f(k)$ to denote this quantity. The best known bounds are that $\frac{15}{14}k -O(1) \le f(k)\le \frac{18}{11}k+O(1)$. The proof of the upper bound is computer-assisted, and considers thousands of cases. We consider $h(k)$, how many prefix and suffix reversals are sufficient to sort any $\pi \in \mathcal{S}_k$. We observe that $\frac{15}{14}k -O(1)\le h(k)$ still holds, and give a human proof that $h(k) \le \frac{3}{2}k +O(1)$. The constant "$\frac{3}{2}$" is a natural barrier for the pancake problem and this variant, hence new techniques will be required to do better.