Symmetry property of \(q\)-weighted Robinson-Schensted algorithms and branching algorithms

In this paper a symmetry property analogous to the well known symmetry property of the normal Robinson-Schensted algorithm has been shown for the \(q\)-weighted Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin. This approach, which uses “growth graphs”, can also be applied to a wider class of insertion algorithms which have a branching structure.

Growth graph of q-RS for 1423

Above is the growth graph of the \(q\)-weighted Robinson-Schensted algorithm for the permutation \({1 2 3 4\choose1 4 2 3}\).