Research

@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}
}

Computing const(p)

To compute const(p), the paper uses the monomial weights $w_j = x^{j-1}$. The sum then reduces to permutations with $\sigma(1) = 1$, which are built from right to left while tracking the exponent of $x$: each weight raises the exponent and the $p$ derivatives lower it. If the exponent would become negative, the term vanishes, and all permutations with that ending are discarded. The remaining permutations form the set $\Phi_p$. Each contributes a signed product of falling factorials, and dividing the total by $0!\,1!\cdots(2p-1)!$ gives const(p). For $p = 5$, only 53,109 of the 3,628,800 permutations remain. Reaching p = 18 requires a different computation, a dynamic programme over subsets carried out modulo primes, which is built up step by step on its own page.

The simulation is limited to p ≤ 5. For p = 6 the set Φ₆ already has 4,605,271 elements; larger cases are handled by the C++ solvers described under Code.

Results so far

The parity theorem.

The paper expresses const(p) as a signed sum over $\Phi_p$, and records as an empirical observation that the numbers of even and odd permutations in $\Phi_p$ differ by exactly one. This is now proved:

$$\sum_{\sigma\in\Phi_p}(-1)^{\sigma} \;=\; (-1)^{p+1} \quad \text{for every } p \ge 1,$$

In particular, $|\Phi_p|$ is odd. The proof encodes each permutation as a lattice path that never goes below zero and constructs two explicit sign-reversing involutions, which pair all permutations except one. The argument does not use primality. The identity has been verified over all of $\Phi_p$ for $p \le 6$ (4,605,271 permutations at $p = 6$) and by dynamic programming up to $p = 19$.

The p-adic conjecture.

The paper proves $v_p(\mathrm{const}(p)) \ge p - 1$ for every prime $p$. Equality holds in all computed cases (the outlined cells in the heatmap below) and is stated in the paper as Conjecture 1. The parity theorem is the key step towards this conjecture: combined with a localisation argument, it reduces the conjecture to a statement that the parity theorem answers. This argument is currently being written up.

Growth.

$\log\mathrm{const}(p)$ grows like $\alpha\,p^2\log p$ with $\alpha$ close to 2. O. Zaboronski (private communication) obtained the bounds

$$p^2\log p\,\bigl(1+O(1/\log p)\bigr) \;\le\; \log\mathrm{const}(p) \;\le\; 2p^2\log p\,\bigl(1+O(1/\log p)\bigr)$$

$$\begin{gathered}p^2\log p\,\bigl(1+O(1/\log p)\bigr)\\ \le\; \log\mathrm{const}(p) \;\le\\ 2p^2\log p\,\bigl(1+O(1/\log p)\bigr)\end{gathered}$$

and the computed values lie close to the upper bound. A least-squares fit of the six-term expansion $\alpha p^2\log p + \beta p^2 + \gamma\,p\log p + \delta p + \varepsilon\log p + \mu$ to the exact values, in 300-digit arithmetic, gives $\alpha = 1.9994 \pm 0.0003$ with all coefficients free. With $\alpha = 2$ fixed, $\beta = -2.6755$, varying by about $10^{-4}$ between fitting windows, and all residuals for $p = 10,\dots,18$ are below $5\times10^{-7}$.

The sequence and the code.

The sequence const(p) is OEIS A392714, with a b-file up to $n = 15$. Exact values are now known up to $p = 18$. They were computed by dynamic programmes over subsets, written in C++, which work modulo primes and reconstruct the result by the Chinese remainder theorem; the number of primes is fixed by a proved upper bound on the size of const(p). The code is publicly available.

A multidimensional Wronskian identity.

Conjecture: the analogous constant equals $1$ in every dimension - confirmed across all tested $(d, k)$.

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.

Code

The solvers are available at kiancshah/wronskian-constant under the MIT licence. Two independently written solvers agree residue by residue, and identity_check.py recovers const(1) = 1 and const(2) = 2 by expanding the operator identity symbolically, without using the recursion. const(18), with 463 digits, was computed on a single node of the Hábrók cluster. How the solvers work is shown step by step on its own page.

Open problems

• The $p$-adic equality. $v_p(\mathrm{const}(p)) = p - 1$: the lower bound is proved and equality holds in all computed cases. The remaining step goes through the parity theorem and is being written up.
• The multidimensional Wronskian conjecture. The analogous constant equals $1$ in every dimension - true in every case tested, proof in progress (intuitively we believe only a few modifications are needed to our proof for the one-dimensional identity).

Details in the paper - feel free to check it out and get in touch.

const(p) also appears in connection with the flag variety and Schubert calculus. I am currently studying this connection, starting from “Chains in the Bruhat order” (arXiv:math/0502363).

Talks & posters

• GQT Graduate School & Colloquium 2026 - Geometry & Quantum Theory cluster, Groesbeek (near Nijmegen), The Netherlands, 24 - 28 August 2026. Talk - slides (PDF)
• Group36 - 36th International Colloquium on Group Theoretical Methods in Physics, Valladolid, Spain, 13 - 17 July 2026. Talk given Friday 17 July - slides (PDF)
• ANTS XVII - 17th Algorithmic Number Theory Symposium, Groningen, The Netherlands, 6 - 10 July 2026. Poster presented: view it as a website ↗ · download the PDF
  (yes, the poster was built in HTML - it lives happily in both formats)
• Honours College - University of Groningen. Talk, June 2026 - slides (PDF)