= &
           \langle {\hat G}^*(\delta_v),\Psi(B_n) \rangle \\           
    & & 
    & = &
           \langle \delta_v,{\hat G}(\Psi(B_n)) \rangle \\  
    & &
     & = &
           \langle \delta_v,\Psi(B_{n+1}) \rangle \\
    & &
     & = &
           \beta(v),
     \end{array}$$ where the second last equality uses the identity ${\hat G}(\Psi(B_n)) = \Psi(B_{n+1})$.

Simple applications {#ss:sa}
-------------------

We now show some interesting consequences of the ideas discussed so far.

\[l:easy\] Let $u$ and $v$ be ${{{\bf c}}{{\bf d}}}$-monomials of degree $m$ and $n$ respectively. Then $$\beta(u \centerdot v) = \binom {m+n+2}{m+1} \beta(u) \beta(v).$$

The key fact to use is that an interval in a Boolean lattice is again a smaller Boolean lattice. The dual to equation  is the identity $$\Psi_P^*(u \centerdot v) = \sum_{{\hat{0}}< x < {\hat{1}}} 
\Psi_{([{\hat{0}},x])}^*(u)  \Psi_{([x,{\hat{1}}])}^*(v),$$ where $\Psi_{P}^*(w)$ denotes the coefficient of $w$ in $\Psi(P)$. Setting $P$ to be the Boolean lattice $B_{m+n+2}$, we obtain $$\beta(u \centerdot v) = \sum_{{\hat{0}}< x < {\hat{1}}} \beta(u) \beta(v),$$ where the sum is over those $x$’s in $B_{m+n+2}$ whose rank is $m+1$. The result now follows.

\[eg:warmup\] To illustrate how Theorem \[t:derivation\] and Lemma \[l:reduction\] work together, we compute $\beta(\underbrace {1 \centerdot 1 \centerdot \ldots \centerdot 1}_{n}).$ First by Lemma \[l:reduction\], we have $\beta(\underbrace {1 \centerdot 1 \centerdot \ldots 