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

Abstract: The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x<sup>{\,p}$ 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 $w_j = x<sup>{j-1}$ reduces the summation to the much smaller set $Φ<em>p \subseteq S</em>{2p-1} \subsetneq S_{2p}$ 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 $c(18) = 4.881\ldots \cdot 10<sup>{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $$\log c(p) \simeq 2p^2\log p -b p² + \overline{o}(p<sup>2)$$ for $p\gg 1$, with $b\geqslant 2.6744$.