这篇论文给出了 L(2,n) 的对称链分解精确计数公式，并用 AlphaEvolve 数据支持超指数增长猜想，与布尔格的最新结果相呼应。
本文研究两种极小格 L(m,n)（m×n 盒子中的分区）和 M(n)（至多 n 的不同部分分区）的对称链分解（SCD）计数问题。作者给出了 L(2,n) 的 #SCD 显式公式，该公式基于排列反演集。对于固定 m>1，他们猜想 #SCD(L(m,n)) 和 #SCD(M(n)) 均超指数增长，该猜想由 Google DeepMind 的进化编码代理 AlphaEvolve 生成的数据支持。文章还证明 Lusztig 对合（evacuation）可扩展为 SCD 上的对合，由此推出 n>2 时 #SCD(M(n)) 为偶数。最后，他们引入与 SCD 等价的 skew tableaux 序列，并探讨了通过 tableau 回避寻找 SCD 的路径。
On the number of symmetric chain decompositions of the minuscule lattices $L(m,n)$ and $M(n)$
We study the problem of enumerating symmetric chain decompositions (SCDs) of the minuscule lattices $L(m,n)$ of partitions in an $m$ by $n$ box and $M(n)$ of partitions into distinct parts at most $n$. We shift the focus from constructing a single SCD to analyzing the global structure of the set of SCDs. Let $\#SCD(P)$ be the number of symmetric chain decompositions of $P$. We give an explicit formula for $\#SCD(L(2,n))$ based on inversion sets of permutations and conjecture that for fixed $m>1$ both $\#SCD(L(m,n))$ and $\#SCD(M(n))$ grow super-exponentially. These conjectures are supported by data produced by AlphaEvolve, an evolutionary coding agent from Google DeepMind, and are in the same vein as a recent paper of Tomon on the growth rate of $\#SCD$ for the Boolean lattice and hypergrid. We make connections with crystal bases and show that the Lusztig involution (evacuation) extends to an involution on SCDs, which we use to show that $\#SCD(M(n))$ is even for $n>2$. We use skew tableaux sequences, which are equivalent to SCDs, and describe a potential path forward for finding SCDs through a notion of tableaux avoidance. We discuss implications of the conjectures for the problem of computing plethysm coefficients and discuss connections to physics and geometry. We end with a list of conjectures, questions and open problems.