The hook length formula gives the number of standard Young tableaux associated with a given Ferrers diagram
with 
boxes. In each box, write its hook length. The number
of standard Young tableaux is then 
divided by the product of all
hook lengths.
More generally, the number of linear extensions of a d-complete poset 
is 
for its recursively defined hook
lengths. This includes rooted trees and standard
shifted Young tableaux as well as ordinary standard Young
tableaux. Nguyen et al. (2026, pp. 7-8) give a purely combinatorial
proof of the multivariate generalization using an extension
of the Robinson-Schensted-Knuth
correspondence.
-Complete Posets." Elec. J. Combin. 33,
P4.1, 2026.