Short permutation pattern waves: The complete length-five classification

Lihong Qiu; Minglei Mao; Yaping Mao

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arXiv · 2611.04184

Short permutation pattern waves: The complete length-five classification

Lihong Qiu·Minglei Mao·Yaping Mao

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Abstract

For a permutation pi in Sk, a pi-wave is an increasing sequence of integers whose consecutive gaps have relative order pi. Let g(pi, n) be the maximum cardinality of a pi-wave-free subset of [n], and let P(pi, r) be the least integer N such that every r-coloring of [N] contains a monochromatic pi-wave. We determine the asymptotic orders of both functions for every permutation of length at most five. Existing upper bounds and weak-wave constructions provide sharp bounds for all but eight length-five patterns in S_5. For these eight exceptional alternating patterns, we establish g(pi, n) = Theta((log n)squared) and P(pi, r) = Theta(r(log r)squared).

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Lihong Qiu, Minglei Mao, Yaping Mao. 2026-11-03. Short permutation pattern waves: The complete length-five classification. https://arxiv.org/abs/2611.04184

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