A Permutation Avoidance Game with Reverse Replies and Monotone Traps

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.

Figures (8)

• Figure 1: Distributions of complete play lengths under optimal play for PAP on $S_n$ with pattern length $4$, for $5 \leq n \leq 10$. Bar heights are $\log_{10}$ of the number of optimal play lines of the given length.
• Figure 2: The eight patterns in $B_6$. Top row: $p_6,q_6,r_6,s_6$. Bottom row: their complements.
• Figure 3: The cell decomposition determined by an increasing pattern $M=\operatorname{id}_4$ when $k=5$. The cell $(i,j)$ records that $i$ points of $M$ lie to the left and $j$ lie below. The shaded cells are the six cells from Lemma \ref{['lem:dangerous-cells']}.
• Figure 4: The three cases in the proof of Theorem \ref{['thm:quadratic-bound']}. The chosen $4$-point subsequence places $x$ in one of the dangerous cells from Lemma \ref{['lem:dangerous-cells']}.
• Figure 5: Logical flow of the $k=4$ reverse-reply proof.

Theorems & Definitions (69)

• Definition 1.1
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof