Linusson-Verkama 线之外的 1324-避免排列的反演单调性
Residual structure and growing inversion-monotonicity regions for 1324-avoiding permutations
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中文总结 AI 辅助
本文研究1324-避免排列的反演计数单调性猜想，通过残差分类与骨架约简，将不等式证明推进至k≤2n+3，比原线多十步。
中文摘要 AI 辅助
设 $a(n,k)$ 为长度为 $n$ 且具有 $k$ 个反演的 $1324$-避免排列的数目。Claesson、Jelinek 和 Steingrimsson 猜想 $a(n,k)\le a(n+1,k)$，这将把该类别的增长率限制在 $13.002$；Linusson 和 Verkama 利用定义在可分解和几乎可分解排列上的注入 $f\sqcup g$ 证明了 $k\le2n-7$ 的情形。我们研究残差 $\mathcal{R}_{\delta,n}$，即缺陷 $\delta=k-2n+7\ge1$ 下不可分解且非几乎可分解的避免者，并证明恒等式 $a(n+1,k)-a(n,k)=|\mathrm{Av}^k_{n+1}(1324)\setminus\mathrm{im}(f\sqcup g)|-|\mathcal{R}_{\delta,n}|$。对缺陷一和缺陷二的残差进行分类，得到 $|\mathcal{R}_{1,n}|=8(n-7)$ 和 $|\mathcal{R}_{2,n}|=32n-214$，这是 $k=2n-6$ 时差值的精确公式，以及所有 $k\le2n-5$ 时的不等式。在此基础上，我们证明了一个骨架约简：$\mathcal{R}_{\delta,n}$ 中的成员资格归结为膨胀块大小的一个二次方程以及仅从骨架读取的非单例条件。在缺陷 $\delta$ 处出现的每个骨架长度至多为 $8\delta+25$；该证明依赖于关于 $1324$-避免者反演图的两个看似新颖的事实。因此，对于大的 $n$，$|\mathcal{R}_{\delta,n}|$ 与次数至多为二的多项式一致，该多项式明确地表示为二项式系数之和。关于割点和反演度为二的条目的四个引理进而表明，每个长度至少为 $23$ 的残差都允许一个可接受的删除；这确定了所有 $\delta\le10$ 时所有 $n$ 的 $\mathcal{R}_{\delta,n}$，并得出对所有 $k\le2n+3$ 的 $a(n,k)\le a(n+1,k)$，比 Linusson-Verkama 线多出十步。该猜想的完整形式，即对所有 $n$ 和 $k$ 成立，仍然开放。
英文摘要
Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\le a(n+1,k)$ for $k\le2n-7$. We study the obstruction beyond that line: the residuals $\mathcal R_{δ,n}$, namely the indecomposable, non-almost-decomposable avoiders at defect $δ=k-2n+7$. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. It follows that, for every fixed $δ$, the eventual count has the form $|\mathcal R_{δ,n}|=A_δn^2+B_δn+C_δ$. Our central uniform result determines the quadratic coefficient at every defect: with $P(q)=\prod_{j\ge1}(1-q^j)^{-1}$, $\sum_{δ\ge0}A_δq^δ=4q^3(1+q)P(q)^2/(1-q)^2$. This is a formula for the leading coefficient of the residual count, not for the full count. The same structural estimates give computer-assisted proofs of $a(n,k)\le a(n+1,k)$ for every $n\ge1$ and $k\le2n+6$, and of regions whose width grows with $n$: for $n\ge2^{16},2^{18},2^{20}$ the defect may be as large as $\lfloor\sqrt n/4\rfloor$, $\lfloor\sqrt n/3\rfloor$, $\lfloor\sqrt n/2\rfloor$, respectively. More generally, every fixed $c<\sqrt2\log(5)/\log(68)$ is admissible for all sufficiently large $n$. The three added fixed defects $11,12,13$ use complete catalogue and rational-sum certificates supplied in the accompanying archival supplement. The leading-coefficient theorem is obtained from a complete finite classification of marked rank-three cores and all-parameter extension lemmas. We also determine the exact rank-three stabilization onset, while keeping it separate from the still unknown onset of the complete residual count. The unrestricted Claesson--Jel'inek--Steingr'imsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.
发表机构
- Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles（加州大学洛杉矶分校地球、行星与空间科学系）
机构由 AI 辅助整理，请以论文原文为准。
