An elementary proof of the hook formula
type: Item
instance of: scholarly article
title: An elementary proof of the hook formula
zbMATH Open document ID: 1179.05118
author: Jason Bandlow
published in: The Electronic Journal of Combinatorics
publication date: 2009-04-07T00:00:00Z
full work available at URL: https://eudml.org/doc/117233
full work available at URL: v15i1r45.html
review text: 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.
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 5540950
zbMATH Keywords: hook length formula
zbMATH Keywords: number of standard tableaux
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5540950