Another Involution Principle-Free Bijective Proof of Stanleyʹs Hook-Content Formula

Another bijective proof of Stanleyʹs hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the authorʹs previous bijective proof (1998, Discrete Math. Theoret. Comput. Sci.3, 011–032) and the Novelli–Pak–Stoyanovskii bijection (J. C. Novelli et al., 1997, Discrete Math. Theoret. Comput. Sci.1, 53–67) for the hook formula for standard Young tableaux of a given shape. This new algorithm can also be used as an algorithm for the random generation of tableaux of a given shape with bounded entries. An appropriate deformation of this algorithm gives an algorithm for the random generation of plane partitions inside a given box.