A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof

The cdebruted ri-we-Rdinson-ThraU (Canad. J. Math. 6 (1954) 316-324) hook-lengths formula, counting the foung tableaux of a specified shape, is given a shart bijective proof. This proof was obtained by translating the elegant Greene-Nijenhuis-Wilf proof (Adv. in Math. 31 (1979) f&l-109) into bijective language. 1 GjG&) satisfying qj <*+lj and ej C= Rj+l (whenever applicable) such that every integer between 1 and n(= AI + l l l + A,,,) appears exactly once among its II entries. For example 12 4 3 6 10 5 7 8 9 is a Young tableau of shape (3,3,2,2). The set of cells ((i, j) : 1 G i c m, 1 ~j < Ai} constituks the shape of the tableau, S(A), and for every cell (i, j) in S(A) we define its hook Hij by Hii = {(a,~)~S(A):a=i and 6a.f or cy 2 i and p = j}. The number of C&S in Hij is denoted by hii.