The object at the center of it all - our paper starts by proving the identity
$$\sum_{\sigma \in S_N} (-1)^\sigma \, w_{\sigma(1)}(x) \cdots w_{\sigma(N)}^{(p)}(x) \;=\; \mathrm{const}(p)\cdot \mathrm{Wronskian}(w_1, \dots, w_N)\cdot f^{(p)}(x),$$which pins the entire left-hand side alternating sum to a single integer constant - and $\mathrm{const}(p)$ turns out to have a life of its own.
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 $w_1, \ldots, w_N$. The case $p = 1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1) = 1$. When $p = 2$, finding $\mathrm{const}(2) = 2$ is easy; we obtain $\mathrm{const}(3) = 90$. The problem is to know $\mathrm{const}(p \geqslant 4)$. We express the formula of $\mathrm{const}(p)$ in terms of the sum with signs over the much smaller set of ‘late-growing’ permutations, thus reaching the exact values $c(p{=}4) = 586\,656$, $c(p{=}5) \approx 1.9 \cdot 10^{12}$, and $c(p{=}6) \approx 7.9 \cdot 10^{21}$; the positive integer sequence $\mathrm{const}(p)$ seems to be new.
Computed exactly with the parallel backtracking algorithm
(const_p_parallel.py); const(14) has 241 digits.
| p | digits | log const(p) | const(p) |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | 1 | 0.6931 | 2 |
| 3 | 2 | 4.4998 | 90 |
| 4 | 6 | 13.2822 | 586656 |
| 5 | 13 | 28.2808 | 1915103977500 |
| 6 | 22 | 50.4194 | 7886133184567796056800 |
| 7 | 35 | 80.4382 | 85873408332103907284746052081828368 |
| 8 | 52 | 118.9567 | 4594491123326092088701002220876785865521537220214784 |
| 9 | 73 | 166.5086 | 2059560342372549855311520646192231857828658156995213197895311773186910208 |
| 10 | 98 | 223.5632 | 12366585616687922175229690518161006581890827598464384763435846689194555321926721635186969600000000 |
| 11 | 127 | 290.5393 | 1512170566890529388470809077236613950878090470607103334738020704327297913201742894343506377843406342016226367608547680866000000 |
| 12 | 160 | 367.8155 | 5498204731735819824932764431577828853091535075963690476426743279778624624056195101600179165694626009665228817515586720358929229075113647839992833114429030400000 |
| 13 | 198 | 455.7371 | 839630392364690485033794364133039282555694250991715962816289209038314705544359344129440199511545622749161083016799037230605187931570735637773841615964174834585398693013887219584559285780585126502400 |
| 14 | 241 | 554.6216 | 7398040169232585459350131400876616433108456563542326671820170467323437649015925253611993741824016395372908844522732645092417947941972247942351664804568139456872774721959807686464898490851546742468976729501727604943676101572611708283345305600 |
@misc{shah2026wronskian,
title = {The alternating compositions of weighted differential
operators yield the weights' {W}ronskian with which
constant?},
author = {Shah, Kian C. and Kiselev, Arthemy V.},
year = {2026},
eprint = {2605.11137},
archivePrefix = {arXiv},
primaryClass = {math.CO},
url = {https://arxiv.org/abs/2605.11137}
}
const_p_parallel.py, GitHub soon) and calculated up till
$p = 14$ - soon live as
OEIS A392714.fit 1: $c\,p\log(p!) + d\log\bigl((2p)!\bigr)$ c = 1.76110 · d = −0.98956 · R² = 0.999989 · max|resid| = 1.07
fit 2: $c\,p^2\log p + d\,p^2$ c = 1.71913 · d = −1.71332 · R² = 0.999967 · max|resid| = 1.94
The longer arc points toward the algebraic structures behind these identities - Lie theory, L∞-algebras and their appearances in mathematical and theoretical physics. That’s the boundary I hope to work at in graduate study: where representation theory, geometry and physics (string-theoretic and otherwise) meet.
Details in the paper - feel free to check it out and get in touch.