A deep dive into the sequence A000219, the number of planar partitions. We explore McMahon’s theorem linking box-restricted planar partitions to lozenge tilings of a hexagon with a hyperfactorial formula, and the recent dual McMahon results counting tilings outside a shamrock core. We’ll see how S-chord hexagons, half-integer hyperfactorials via the gamma function, and Kuo’s graphical condensation come together, and how these tilings connect to posets, symmetry classes, and 3D stepped-surface interpretations in the OEIS web of ideas.

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