发表机构
Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen（格罗宁根大学伯努利数学、计算机科学与人工智能研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究加权微分算子交替复合的Wronskian系数c(p)，通过晚期增长排列将求和从(2p)!缩减至更小集合，实现快速计算并得到新序列OEIS A392714，最高计算至c(18)≈4.881×10^462，并给出其增长率渐近形式。
AI 中文摘要
在实数线 $\mathbb{R} \ni x$ 上，$N = 2p$ 个严格阶为 $p$ 的加权微分算子 $w_j(x)\cdot\partial_x^{\\,p}$ 的交替复合仍然是阶为 $p$ 的算子；其系数是普适常数 $c(p)$ 乘以权重 $w_1,\ldots,w_N$ 的 Wronskian。向量场的李括号固定 $c(p=1)=1$；我们想要找到 $c(p \geqslant 2)$：例如，$c(2) = 2$ 或 $c(3) = 90$。直接符号展开（涉及 $|S_{2p}| =(2p)!$ 个排列）在 $p \geqslant 4$ 时失败。取单项式 $w_j = x^{j-1}$ 将求和缩减到更小的集合 $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$，即晚期增长排列。将 $c(p)$ 表示为下降阶乘乘积的带符号和，我们实现并加速了算法，该算法获得了直到 $c(18) = 4.881\ldots \cdot 10^{462}$ 的所有整数值。所得序列是新的，现已注册为 OEIS A392714；其（次）主导阶增长率为 $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$（当 $p\gg 1$ 时），其中 $b\geqslant 2.6744$。
英文摘要
The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $Φ_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.
CommentsExpanded extract from arXiv:2605.11137 [math.CO]; contains a new 206-decimal-digit confirmed prime; 30 pages, 9 tables, 1 figure, 3 appendices; program code in Appendix B and on github (external)
