Gröbner bases of radical Li-Li type ideals associated with partitions

For a partition $\lambda$ of $n$, the Specht ideal $I_{\lambda }\subset K[x_1,\dots ,x_n]$ is the ideal generated by all Specht polynomials of shape $\lambda$. In their unpublished manuscript, Haiman and Woo showed that $I_{\lambda }$ is a radical ideal, and gave its universal Gröbner bases (recently, Murai et al. published a quick proof of this result). On the other hand, an old paper of Li and Li studied analogous ideals, while their ideals are not always radical. In the present paper, we introduce a class of ideals which generalizes both Specht ideals and radical Li-Li ideals, and study their radicalness and Gröbner bases.