= Staircase partition has exactly the odd-hook property
{title2=$\lambda=(k,k-1,\ldots,1)\iff\text{every hook length is odd}$}
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.
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