A `nice' bijection for a content formula for skew semistandard Young tableaux (Q5960783)

An exercise in the book [\textit{R. P. Stanley}, Enumerative combinatorics. Volume 2 (Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge) (1999; Zbl 0928.05001)] asks for a `nice' bijective proof of an identity first established in [\textit{S. C. Billey, W. Jockusch} and \textit{R. P. Stanley}, J. Algebr. Comb. 2, 345-374 (1993; Zbl 0790.05093)]. For a fixed shape \(\lambda/\mu\) this identity relates the generating functions \(\sum q^{n(P)}\) and \(\sum q^{n(R)}\), where \(n(P)\) is the number of all semistandard Young tableaux, \(n(R)\) is the number of all reverse semistandard Young tableaux with entries \(R_{ij}\leq a+\mu_i-i\) and \(a\) is an arbitrary integer such that \(a+j-i>0\) for all cells \((i,j)\) of \(\lambda/\mu\). In the paper under review the author presents such a bijective proof based on algorithms which combine modifications of the Schützenberger evaluation and of jeu de taquin. The paper contains also a step-by-step example of the work of the algorithms and a table giving a complete match between different kinds of tableaux involved in the algorithms.