A direct bijective proof of the hook-length formula

The aim of this paper is to give a bijective proof of the hook-length formula for the enumeration of standard Young tableaux of a given shape. This formula was discovered by Frame, Robinson and Thrall in 1954 [1] and since then it has been the object of much study. Many proofs have been published based on different approaches, but none of them is considered satisfactory. We refer to the paper [9] for a well written review of the different proofs and their history, but we want to recall a few of them and some related papers which had a strong impact on us (see also the references in [5, 10]). First, we should mention the remarkable paper of Schützenberger [11]. It is not directly related to the hook-length formula but contains the famous jeu de taquin algorithm. Our bijection is based on this procedure. The first major steps in the understanding of the hook-length formula were made by Hillman and Grassl [4] and Stanley [12]. Their two beautiful bijections combined give the result, although an algebraic step is also needed. yUniversité Paris 7 – LIAFA – 2, place Jussieu 75251 Paris Cedex 05 – email: [email protected] zDepartment of Mathematics – Harvard University – Cambridge MA 02138. email: [email protected]. xDepartment of Mathematics – Independent University – Moscow Russia. email: [email protected].