Generalization of Schensted insertion algorithm to the cases of hooks and semi-shuffles

Abstract

Given an rc-graph $R$ of permutation $w$ and an rc-graph $Y$ of permutation $v$, we provide an insertion algorithm, which defines an rc-graph $R\leftarrow Y$ in the case when $v$ is a shuffle with the descent at $r$ and $w$ has no descents greater than $r$ or in the case when $v$ is a shuffle, whose shape is a hook. This algorithm gives a combinatorial rule for computing the generalized Littlewood-Richardson coefficients $c^{u}_{wv}$ in the two cases mentioned above.