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

Given an rc-graph R of permutation w and an rc-graph Y of a permutation v, we provide an insertion algorithm which defines an rc-graph R←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 cwvu in the two cases mentioned above.