A geometric version of the Robinson-Schensted correspondence for skew oscillating tableaux
The Robinson-Schensted correspondence is a classical combinatorial construction relating permutations and pairs of standard tableaux having the same shape. More recently, people have been interested in the extension of this correspondence to various generalizations, in the Young poset, skew and generalized skew tableaux, shifted tableaux, etc. In the paper under review the authors consider the extension of the Robinson-Schensted correspondence to the family of skew oscillating tableaux introduced first by T. Roby in his Ph.D. Thesis, M.I.T., 1991; see [\textit{T. Roby}, Discrete Math. 139, 481-485 (1995; Zbl 0920.05075)] and later independently in [\textit{S. Dulucq, B. E. Sagan}, Discrete Math. 139, 129-142 (1995; Zbl 0843.05101)]. The authors define an analogue of the geometric construction of [\textit{G. Viennot}, Comb. Represent. Groupe symetr., Actes Table Ronde C. N. R. S. Strasbourg 1976, Lect. Notes Math. 579, 29-58 (1977; Zbl 0389.05016)] and deduce new geometric (and intuitive) proofs of some combinatorial properties of this correspondence. In particular, the authors establish a new result about the number of odd height columns in the final shape of a skew oscillating tableau, which extends similar results for standard tableaux [\textit{M.-P. Schützenberger}, Comb. Represent. Groupe symetr., Actes Table Ronde C. N. R. S. Strasbourg 1976, Lect. Notes Math. 579, 59-113 (1977; Zbl 0398.05011)], [\textit{J. S. Beissinger}, Discrete Math. 67, 149-163 (1987; Zbl 0649.05002)] and skew tableaux [\textit{B. E. Sagan} and \textit{R. P. Stanley}, J. Comb. Theory, Ser. A 55, 161-193 (1990; Zbl 0732.05061)].
scientific article; zbMATH DE number 1741702