A symmetry property for $$q$$ q -weighted Robinson–Schensted and other branching insertion algorithms

Abstract

In [19], a $$q$$ q -weighted version of the Robinson---Schensted algorithm was introduced. In this paper, we show that this algorithm has a symmetry property analogous to the well-known symmetry property of the usual Robinson---Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin [5---8]. This approach, which uses `growth graphs', can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other $$q$$ q -weighted versions of the Robinson---Schensted algorithm which have recently been introduced by Borodin---Petrov [2].