Palindromicity of Antichain Polynomials of Three-Dimensional Boxes

Abstract

For the antichain generating polynomial of the product poset [a] x [b] x [c], this preprint gives a complete palindromicity classification. After sorting a ≤ b ≤ c, the polynomial is palindromic exactly for (a,b,c)=(1,r,r) or (2,r,r+1). The new higher-layer theorem excludes every box with a,b,c ≥ 3 through exact next-to-leading coefficients derived from last-passage matrices, path complementation, and minimum separators. Combined with the known one- and two-layer cases, this settles the rectangular minuscule branch of Ding and Dong's Conjecture B. Status: Public Beta; internally verified candidate proof; external mathematical review pending.