New integer sequence OEIS A392714 counts Wronskians: fast evaluation via late-growing permutations

Abstract

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 $w_1,\dots ,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p⩾2)$: e.g., $c(2)=2$ or $c(3)=90$. Direct symbolic expansion (over $|S_{2p}|=(2p)!$ permutations) fails for $p⩾4$. Taking the monomials $w_j=x^{j-1}$ reduces the summation to the much smaller set ${\Phi }_p\subseteq S_{2p-1}⊊S_{2p}$ 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 $c(18)=4.881\dots \cdot 10^{462}$. 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 $b⩾2.6744$.

arXiv commentExpanded 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)