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, $\\textbf{ Z}\\st{n}* \\textbf{ Z}\\st{n}*\\ldots \\st{n}*\\textbf{ Z}$) 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 $(c_1,c_2,\\ldots,c_t)$, where $c_1\\geq n$.

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, $\\textbf{ Z}\\st{n}* \\textbf{ Z}\\st{n}*\\ldots \\st{n}*\\textbf{ Z}$) 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 $(c_1,c_2,\\ldots,c_t)$, where $c_1\\geq n$.

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