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

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, ω)-partition theorem with the in-volution principle of Garsia and Milne. 1. Introduction. A few years ago there had been a lot of interest in finding a bijective proof of Frame, Robinson and Thrall's [1] hook formula for the number of standard Young tableaux of a given shape. This resulted in the discovery of three different such proofs [2, 10, 14], none of them is considered to be really satisfactory. Closest to being satisfactory is probably the proof by Franzblau and Zeilberger [2]. However, while the description of their algorithm is fairly simple, it is rather difficult to show that it really works. Also, it does not portray the nice row-column symmetry of the hooks. Remmel's proof [10] is the most complicated. It uses the involution principle of Garsia and Milne [3]. However, Remmel bases his proof on " bijectivization " of recurrence relations, which is not the most direct route to attack the problem. Finally, Zeilberger's proof [14], translating the beautiful probabilistic proof [6] by Greene, Nijenhuis and Wilf into a bijection, actually sets up a bijection between larger sets than one desires. So, it is still considered to be the case that the best proof of the hook formula is to use the Hillman–Grassl algorithm [7] and Stanley's (P, ω)-partition theorem [12], and then to apply a limit argument (this is the non-bijective part). In view of this it is somehow surprising that there are half-combinatorial proofs of the hook formula