On mixed insertion, symmetry, and shifted young tableaux

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Abstract

We introduce a generalization, mixed insertion, of Schensted insertion and develop certain remarkable relationships between mixed insertion and an operation of conversion which generalized Schützenberger's jeu-de-taquin. Applying results about mixed insertion to symmetric tableaux, we answer several open questions relating to a shifted analog of Schensted insertion studied by Worley and Sagan in connection with the Hall-Littlewood symmetric functions Qλ(x; − 1). The questions answered include the effect on shifted insertion of taking the inverse of a permutation, the effect of reversing the ordering of the alphabet of symbols, a conjecture of Shor relating shifted and unshifted insertion, and a fuller explication of the evacuation operator.

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