Enumeration formulas for standard Young tableaux of nearly hollow rectangular shapes

This paper considers the enumeration problem of a generalization of standard Young tableau (SYT) of truncated shape. Let $\lambda ?\mu |\left\{(i_0,j_0)\right\}$ be the SYT of shape $\lambda$ truncated by $\mu$ whose upper left cell is $(i_0,j_0)$, where $\lambda$ and $\mu$ are partitions of integers. The summation representation of the number of SYT of the truncated shape $(n+k+2,{(n+2)}^{m+1})?\left(n^m\right)|\left\{(2,2)\right\}$ is derived. Consequently, three closed formulas for SYT of hollow shapes are obtained, including the cases of (i). $m=n=1$, (ii). $k=0$, and (iii). $k=1,m=n$. Finally, an open problem is posed.