Hook Length

The hook length of a box in a Ferrers diagram is one plus the number of boxes to its right in the same row and below it in the same column. Those boxes together with the original box form its hook. The hook lengths determine the number of standard Young tableaux of a given shape through the hook length formula (Skiena 1990).

More generally, let be a finite d-complete poset. Its generalized hook lengths are defined recursively as follows. For , a -interval has a unique incomparable pair, called its side elements. If is the maximum of such an interval with side elements , define

The interval is unique. If no such interval has maximum , define

These generalized hook lengths enter the hook length formula (Nguyen et al. 2026, pp. 6-7, 10).