Bijective proofs of the hook formulas for the number of standard Young tableaux, ordinary and shifted

Abstract

Bijective proofs of the hook formulas for the number of ordinary standard Young tableaux and for the number of shifted standard Young tableaux are given. They are formulated in a uniform manner, and in fact prove $q$-analogues of the ordinary and shifted hook formulas. The proofs proceed by combining the ordinary, respectively shifted, Hillman–Grassl algorithm and Stanley's $(P,\omega)$-partition theorem with the involution principle of Garsia and Milne.