We present an explicit structure for the Baer invariant of a free $n$th nilpotent group (the $n$th nilpotent product of infinite cyclic groups, $\textbf{ Z}\st{ n}* \textbf{ Z}\st{ n}*...\st{ n}*\textbf{ Z}$) with respect to the variety ${ \cal V}$ with the set of words $V=\{ [\ga_{ c_1+1},\ga_{ c_2+1}]\}$, for all $c_1\geq c_2$ and $2c_2-c_1>2n-2$. Also, an explicit formula for the polynilpotent multiplier of a free $n$th nilpotent group is given for any class row $(c_1,c_2,...,c_t)$, where $c_1\geq n$.
Categories: math.GR