这是一份exeter埃克塞特大学MTH3021RWA作业代写的成功案

The definition of boxed products implies the coefficient relation
$$
A_{n}=\sum_{k=1}^{n}\left(\begin{array}{l}
n-1 \
k-1
\end{array}\right) B_{k} C_{n-k} .
$$
The binomial coefficient that appears in the standard labelled product is now modified since only $n-1$ labels need to be distributed between the two components, $k-1$ going to the $\mathcal{B}$ component (that is constrained to contain the label 1 already) and $n-k$ to the $\mathcal{C}$ component. From the equivalent form
$$
\frac{A_{n}}{n !}=\frac{1}{n} \sum_{k=0}^{n}\left(\begin{array}{l}
n \
k
\end{array}\right)\left(k B_{k}\right) C_{n-k}
$$

MTH3021RWA COURSE NOTES ：

A useful special case is the min-rooting operation,
$$
\mathcal{A}={1}^{0} \star \mathcal{C} \text {. }
$$
for which a variant definition goes as follows. Take in all possible ways elements $\gamma \in \mathcal{C}$, prepend an atom with a label smaller than the labels of $\gamma$, for instance 0 , and relabel in the canonical way over $[1 \ldots(n+1)]$ by shifting all label values by 1 . Clearly $A_{n+1}=C_{n}$ which yields
$$
A(z)=\int_{0}^{z} C(t) d t
$$
a result also consistent with the general formula of boxed products.
For some applications, it is easier to impose constraints on the maximal label rather than the minimum. The max-boxed product written
$$
A=\left(B^{-} * \mathcal{C}\right) \text {. }
$$