Formula for nilpotent multiplier of free nilpotent group

Description as a lower central series quotient
Suppose $$c_1,c_2$$ are (possibly equal, possibly distinct) natural numbers.

Suppose $$G$$ is the free nilpotent group of class $$c_1$$ on a generating set $$S$$. If $$F$$ is the free group on $$S$$, $$G$$ is explicitly the quotient group $$F/\gamma_{c_1+1}(F)$$, where $$\gamma_{c_1+1}(F)$$ is the $$(c_1 + 1)^{th}$$ member of the lower central series of $$F$$.

Then, the $$c_2$$-nilpotent multiplier of $$G$$ is given as follows:

$$M^{(c_2)}(G) = \gamma_{\max\{c_1,c_2\} + 1}(F)/\gamma_{c_1 + c_2 + 1}(F)$$

Note that this is an abelian group (as can be seen, from instance, by noting that second half of lower central series of nilpotent group comprises abelian groups). This makes sense because nilpotent multipliers are supposed to be abelian groups.

Explicit formula for dimension
Suppose $$G$$ is a free nilpotent group of class $$c_1 \ge 1$$ on $$n$$ generators. Then, the nilpotent multiplier $$M^{(c_2)}(G)$$ is a free abelian group whose rank is given by summing up the values obtained using the formula for dimension of graded component of free Lie algebra for $$\max\{c_1,c_2 \} + 1 \le r \le c_1 + c_2 + 1$$. Explicitly, it is the double sum:

$$\sum_{r = \max\{ c_1,c_2 \} + 1}^{c_1 + c_2} \left(\frac{1}{r} \sum_{d|r} \mu(d)n^{r/d}\right)$$

For fixed $$c_1$$ and $$c_2$$, this is a polynomial of degree $$c_1 + c_2$$ in $$n$$.

Case $$c_2 \le c_1$$
In this case, the class $$c_2$$-covering group of the free class $$c_1$$-nilpotent group on a generating set $$S$$ is the free nilpotent group of class $$c_1 + c_2$$ on the same generating set $$S$$. Explicitly, if $$F$$ is the free group on $$S$$, then:


 * The group $$G$$ is $$F/\gamma_{c_1 + 1}(F)$$
 * The $$c_2$$-nilpotent multiplier of $$G$$ is $$M^{c_2}(G) = \gamma_{c_1 + 1}(F)/\gamma_{c_1 + c_2 + 1}(F)$$
 * The covering group is $$F/\gamma_{c_1 + c_2 + 1}(F)$$

The short exact sequence is:

$$0 \to \gamma_{c_1 + 1}(F)/\gamma_{c_1 + c_2 + 1}(F) \to F/\gamma_{c_1 + c_2 + 1}(F) \to F/\gamma_{c_1 + 1}(F)\to 1$$

Case $$c_1 \le c_2$$
Does a covering group exist under these circumstances?

Particular cases
The rows here correspond to particular choices of $$c_1$$ and $$c_2$$.