Symmetric chain decompositions of the minuscule lattice M(n): twenty-two instances and two exact counts

Abstract

Let M(n) be the poset of strict integer partitions with largest part at most n, ordered by containment of Young diagrams. Stanley asked in 1980 whether M(n) admits a symmetric chain decomposition; the question is open, and the most recent account, by Dorward, reports that little progress has been made. We exhibit an explicit symmetric chain decomposition of M(n) for every n ≤ 21; the largest splits the 2 097 152 elements of M(21) into 28 460 saturated chains, each symmetric about the middle of the rank range 0,…,231. The decompositions are produced by a middle-out construction in which one outward step is a unit-capacity maximum flow. We also compute two counts for which only lower bounds were known: nscd(M(6))=1344, so that the published bound ≥ 1344 is tight, and nscd(M(7))=295 069 056, about 819 times the published bound ≥ 360 036. The second count is out of reach of exhaustive enumeration and is obtained from a dynamic program whose state is the bijection between two opposite ranks realised by the chains still under construction. All statements are machine-checked in Lean 4.

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