Four fault-free $B_{n-2}$'s in $B_{n}$ under the random node fault model

Let $n\geq 4$. Each $B_{n-2}$ in $B_n$ has one of the forms $a_1a_2X^{n-2}$, $a_1X^{n-2}a_2$ and $X^{n-2}a_1a_2$. Let $1-p$ be the fault probiability of each node in the $n$-dimensional bubble-sort network $B_{n}$ under the random node fault model. In this paper, we determine the probability that there are four distinct fault-free $B_{n-2}$'s in $B_{n}$ by considering all possible combinatorial cases of the four fault-free $B_{n-2}$'s.