We investigate the random permutation matrices induced by the Chinese restaurant processes with $(α,θ)$-seating. When $α=0,θ>0$, the permutations are those following Ewens measures on symmetric groups, and have been extensively studied in the literature. Here, we consider $α\in(0,1)$ and $θ>-α$. In an accompanying paper, a functional central limit theorem is established for partial sum of weighted cycle counts in the form of $\sum_{j=1}^n a_jC_{n,j}$, where $C_{n,j}$ is the number of $j$-cycles of the permutation matrix of size $n$. Two applications are presented. One is on linear statistics of the spectrum, and the other is on the characteristic polynomials outside the unit circle.