Generalized Burnside Algebra of type $B_{n}$

In this paper, we firstly give an alternative method to determine the size of $C(S_{n})$ which is the set of elements of type $S_{n}$ in a finite Coxeter system $(W_{n},S_{n})$ of type $B_{n}$. We also show that all cuspidal classes of $W_{n}$ are actually the conjugate classes $\mathcal{K}_{\lambda}$ for every $\lambda \in \mathcal{DP}^{+}(n)$. We then define the generalized Burnside algebra $HB(W_{n})$ for $W_{n}$ and construct a surjective algebra morphism between $HB(W_{n})$ and Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$. We obtain a set of orthogonal primitive idempotents $e_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $HB(W_{n})$, that is, all the characteristic class functions of $W_{n}$. Finally, we give an effective formula to compute the number of elements of all the conjugate classes $\mathcal{K}_{\lambda}$, $\lambda \in \mathcal{DP}(n)$ of $W_{n}$.