Research

questions worth chasing

Up to normalisation, the constants $c(p)$ appear as the balanced coefficients of the Postnikov–Stanley polynomial $\mathfrak{D}_{w_0}$ for $S_{2p}$, which is a sum over maximal chains in the Bruhat order [Postnikov–Stanley]Chains in the Bruhat orderA. Postnikov, R. P. StanleyJ. Algebraic Combin. 29 (2009)arXiv:math/0502363. We are pursuing this direction of research with Oleg Zaboronski. Below are the results and algorithms established so far.

Late-growing permutations

walks that never go underground
preprint · oct 2026 · math.CO · with Arthemy Kiselev

The structure of late-growing permutations

The permutations behind the Wronskian constant $c(p)$, drawn as walks that never go below zero. We prove that their even and odd counts differ by exactly $(-1)^{p+1}$. Keeping only the walks pinned to the ground at one step still gives $\pm c(p)$, verified for $p \leqslant 14$. Every walk for $p \leqslant 4$ to click through.

step inside → arXiv:2610.35779 ↗ 
@misc{shah2026parity,
  title         = {The counts of parity-odd vs even late-growing
                   permutations in the group $\mathbb{S}_{2p-1}$
                   differ by $(-1)^p$},
  author        = {Shah, Kian C. and Kiselev, Arthemy V.},
  year          = {2026},
  eprint        = {2610.35779},
  archivePrefix = {arXiv},
  primaryClass  = {math.CO},
  url           = {https://arxiv.org/abs/2610.35779}
}

const(p) and Wronskians

one integer per order
preprints · v1 may 2026, v2 july 2026 · math.CO, oct 2026 · math.NT

Wronskian identity for iterated differential operators

Sum $2p$ differential operators $w_j\partial_x^{\,p}$ over all orderings, with signs, and what comes out is the Wronskian of the weights times an integer:

$$\begin{aligned} &\sum_{\sigma \in S_{2p}} (-1)^{\sigma}\, \bigl(w_{\sigma(1)}\partial_x^{\,p}\bigr) \circ \cdots \circ \bigl(w_{\sigma(2p)}\partial_x^{\,p}\bigr)\\ &\qquad=\; \operatorname{const}(p)\cdot \operatorname{Wr}(w_1,\dots,w_{2p})\cdot \partial_x^{\,p}. \end{aligned}$$

const(p) runs 1, 2, 90, 586656, … and we computed it exactly up to p = 18, which has 463 digits.

9 calls 6 states
step inside → arXiv:2605.11137 ↗ arXiv:2610.08636 ↗ 
@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}
}
@misc{shah2026oeis,
  title         = {New integer sequence {OEIS} {A392714} counts
                   {W}ronskians: fast evaluation via late-growing
                   permutations},
  author        = {Shah, Kian C. and Kiselev, Arthemy V.},
  year          = {2026},
  eprint        = {2610.08636},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  url           = {https://arxiv.org/abs/2610.08636}
}

Minimal surfaces

soap films grown from symmetries
sep 2025 - feb 2026 · arXiv:math/0603424

Minimal surfaces from symmetries

The surfaces from Arthemy Kiselev's 2006 paper in 3D, how they're generated, and one whose edge is a trefoil.

step inside →