The answer is the same as the number of ways to select four integers from $\mathbb{S}=\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$ when duplication is permitted. This is because for every such selection, there is a unique way to arrange these four numbers in and increasing order. On the other hand, all qualifying $x_1$, $x_2$, $x_3$ and $x_4$ must be selected from $\mathbb{S}$.

Meanwhile, the number of ways to select $4$ elements from $\mathbb{S}$, duplication permitted, is equivalent to the number of non-negative integer solutions to the following equation where $k_i$ ($i=1,\ 2,\ \cdots,\ 6$) indicates the number of $i$ selected: $$k_1+k_2+k_3+k_4+k_5+k_6=4$$

Therefore the answer is $$\binom{4+6-1}{4}=\boxed{126}$$