An elementary proof of the hook formula
Summary: The hook-length formula is a well known result expressing the number of standard tableaux of shape \(\lambda\) in terms of the lengths of the hooks in the diagram of \(\lambda\). Many proofs of this fact have been given, of varying complexity. We present here an elementary new proof which uses nothing more than the fundamental theorem of algebra. This proof was suggested by a \(q,t\)-analog of the hook formula given by \textit{A. M. Garsia} and \textit{G. Tesler} [``Plethystic formulas for Macdonald \(q,t\)-Kostka coefficients'', Adv. Math. 123, No.\,2, 144--222 (1996; Zbl 0865.05075)], and is roughly based on the inductive approach of \textit{C. Greene}, \textit{A. Nijenhuis}, and \textit{H. Wilf} [``A probabilistic proof of a formula for the number of Young tableaux of a given shape'', Adv. Math. 31, 104--109 (1979; Zbl 0398.05008)]. We also prove the hook formula in the case of shifted Young tableaux using the same technique.
scientific article; zbMATH DE number 5540950