A Permutation Avoidance Game with Reverse Replies and Monotone Traps

TL;DR

This paper investigates a permutation avoidance game, identifying minimal pattern sets that enforce monotonicity, establishing thresholds, and analyzing winning strategies for specific pattern lengths.

Contribution

It introduces the set $B_k$ as the unique minimal monotone-forcing subset of $S_k$, determines thresholds for pattern lengths, and analyzes winning strategies for the game PAP.

Findings

Identified the set $B_k$ as the minimal monotone-forcing subset of $S_k$.

Established quadratic upper bounds for the monotone-forcing threshold.

Proved a reverse-reply strategy wins for $k=4$ and analyzed strategies for $k=3$.

Abstract

We study the impartial game PAP (``permutations avoiding patterns''), in which players take turns choosing patterns to avoid. We define a set of length $k$ patterns, $B_k$, and show that it is the unique minimal monotone-forcing subset of $S_k$: every sufficiently long permutation that avoids $B_k$ is monotone, and every monotone-forcing subset of $S_k$ must contain $B_k$. We prove a quadratic upper bound for the monotone-forcing threshold, and determine the exact thresholds for $k=3,4,5,6$. We use properties of the sets $B_k$ to prove that a reverse-reply strategy wins PAP on $S_n$ when $k=4$ for all $n \geq 10$; for $k=3$, the same strategy can be analysed directly. We conjecture that it is a winning strategy for all $k$ and $n$ sufficiently large.