A Permutation Avoidance Game with Reverse Replies and Monotone Traps
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915868296871936 |
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| author | Ulfarsson, Henning |
| author_facet | Ulfarsson, Henning |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16004 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Permutation Avoidance Game with Reverse Replies and Monotone Traps Ulfarsson, Henning Combinatorics Discrete Mathematics 05A05, 91A46 G.2.1; F.1.3 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. |
| title | A Permutation Avoidance Game with Reverse Replies and Monotone Traps |
| topic | Combinatorics Discrete Mathematics 05A05, 91A46 G.2.1; F.1.3 |
| url | https://arxiv.org/abs/2603.16004 |