Kian C. Shah | Search arXiv

arXiv2026

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

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 $b\geqslant 2.6744$.

math.NT↗

arXiv2026

The alternating compositions of weighted differential operators yield the weights' Wronskian with which constant?

The alternated composition of $N=2p$ differential operators $w_j(x)\,\partial_x^p$ of strict order $p$ on the line $\mathbb{R}\ni x$ is again a differential operator of strict order $p$; its coefficient is the constant $\mathrm{const}(p)$, depending only on the arity $N$, times the Wronskian determinant of the originally taken coefficients $w_1,\dots,w_N$. The case $p=1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1)=1$, and $\mathrm{const}(2)=2$ is found easily by hand; $\mathrm{const}(3)=90$ can still be obtained symbolically. The problem is to determine $\mathrm{const}(p\geqslant4)$. We compute $\mathrm{const}(p)$ exactly for all $p\leqslant14$ -- a 241-digit integer at $p=14$ -- and record the resulting integer sequence as OEIS A392714. We prove that $v_p(\mathrm{const}(p))\geqslant p-1$ for every prime $p$, matching the exact equality observed numerically throughout our range, and conjecture that this equality holds in general. We show that $\log\mathrm{const}(p)$ grows like $αp^2\log p$, with the leading coefficient close to $2$, nearly saturating the bound $p^2\log p\,(1+O(1/\log p))\leqslant\log\mathrm{const}(p)\leqslant2p^2\log p\,(1+O(1/\log p))$ obtained by O. Zaboronski (private communication). A naturally arising reduced constant is found to decay to zero super-exponentially rather than grow, a direct consequence of this near-saturation.

math.CO↗