Regularity of Cohen-Macaulay Specht ideals

For a partition $\lambda$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. In the previous paper, the second author showed that if $R/I^{\rm Sp}_\lambda$ is Cohen-Macaulay, then $\lambda$ is either $(n-d,1,\ldots,1),(n-d,d)$, or $(d,d,1)$, and the converse is true if ${\rm char}(K)=0$. In this paper, we compute the Hilbert series of $R/I^{\rm Sp}_\lambda$ for $\lambda=(n-d,d)$ or $(d,d,1)$. Hence, we get the Castelnuovo-Mumford regularity of $R/I^{\rm Sp}_\lambda$, when it is Cohen-Macaulay. In particular, $I^{\rm Sp}_{(d,d,1)}$ has a $(d+2)$-linear resolution in the Cohen-Macaulay case.