Staircase partition has exactly the odd-hook property ($\lambda =(k,k-1,\dots ,1)⟺\text{every hook length is odd}$)

A partition of an integer has only odd hook lengths exactly when it is a staircase. Absence of a hook of length two forces the last row to have length one, prohibits equal consecutive rows by examining the last boxes of an equal-length block, and prohibits gaps at least two by examining a penultimate row box. Thus consecutive row lengths differ by one. Conversely a staircase has $h(i,j)=2k-2i-2j+3$, always odd. Its size is $k(k+1)/2$. The hook-length formula then gives $v_2(f^{\lambda })=v_2(n!)$, and no other partition of the same size has that full two-adic divisibility, since it would also have an odd hook product.