Hook partition ($(n-k,1^k)$)

A hook partition has shape $(n-k,1^k)$, $0\le k\le n-1$. Equivalently its Young diagram has no cell $(2,2)$, so its only cell of content zero is $(1,1)$. Its standard Young tableaux are counted by $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$: choose the $k$ entries below the initial $1$, then the column and remaining row orders are forced.