On the number of symmetric chain decompositions of the minuscule lattices $L(m,n)$ and $M(n)$

Abstract

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.