Mixing times of one-sided $k$-transposition shuffles

We study mixing times of the one-sided $k$-transposition shuffle. We prove that this shuffle mixes relatively slowly, even for $k$ big. Using the recent ``lifting eigenvectors’’ technique of Dieker and Saliola and applying the ${\mathrm{\ell }}^2$ bound, we prove different mixing behaviors and explore the occurrence of cutoff depending on $k$.