Properties of the Robinson-schensted Correspondence for Oscillating and Skew Oscillating Tableaux

In this paper, we consider the Robinson-Schensted correspondence for oscillating tab-leaux and skew oscillating tableaux deened in 15] and 3]. First we give an analogue, for the oscillating tableaux, of the classical geometric construction of Viennot for standard tableaux ((16]). Then, we extend a construction of Sagan and Stanley ((10]), dealing with standard tableaux and skew tableaux, to deduce a property about the number of odd height columns of a skew oscillating tableau. Finally, we give analogues for skew oscillating tableaux of two classical constructions about this correspondence, Knuth classes ((7]) and Beissinger algorithm ((1]). 1. Introduction The Robinson-Schensted correspondence is a classical bijection between permutations and pairs of standard tableaux of the same shape. It was deened in 11], and followed by numerous papers dealing with the combinatorial properties of this correspondence, like 12], 7], 5], 13], 16] or 1]. More recently, this correspondence was extended to various kinds of tableaux that are generalizations, in the Young lattice, of the standard tableaux: semi-standard tableaux ((7]), skew tableaux ((10]), oscillating tableaux ((rst by Sundaram in 14, 15], then independtly Delest, Dulucq and Favreau in 2]) and skew oscillating tableaux ((3]). In this article, we extend classical properties and constructions related to the Robinson-Schensted correspondence to the correspondences for oscillating tableaux and skew oscillating tableaux. In the sections 2 and 3, we give basic deenitions on biwords and tableaux and we present the correspondence for skew oscillating tableaux. Then we extend a geometric version of the Robinson-Schensted correspondence due to Viennot ((16]) to the case of oscillating tableaux, and, following 10], we deene a construction allowing extension of properties of oscillating tableaux to skew oscillating tableaux. Finally, in sections 6 and 7, we extend to skew oscillating tableaux a result of Knuth ((7]) and an algorithm of Beissinger ((1]).