Regularity of Cohen-Macaulay Specht ideals

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