d-Complete Poset

A -complete poset is a finite partially ordered set satisfying local completion and nonoverlap axioms for double-tailed diamonds. For each , let consist of two incomparable elements, a chain of elements below both, and a chain of elements above both. An order interval is a -interval when it is isomorphic to . A -convex subset is a copy of with its maximum removed that contains every element lying between two of its elements. The axioms require that every such subset can be completed to a -interval, that the maximum of a -interval covers no element outside the interval, and that no two -convex subsets differ only in their minimal elements (Nguyen et al. 2026, Definition 4).

Rooted trees and the partial orders associated with ordinary and shifted Young tableaux give examples with the order conventions of Nguyen et al. (2026). The number of linear extensions of a -complete poset satisfies the hook length formula

where the hook length is defined recursively. If is not a neck element (one lying above the two incomparable sides) of any double-tailed diamond interval, then is the number of elements at or below . If is the maximum of such an interval with incomparable side elements , then . Nguyen et al. (2026) give a purely combinatorial proof of a multivariate version using a generalization of the Robinson-Schensted-Knuth correspondence.