Partitions of the interval $[1,n]$ and the group action of $\mathfrak{S}_{n}$

Question

Let $r,n\in\mathbb{N}_{\geq2}$ and $\mathfrak{S}_{n}$ be the symmetric group of the interval $[1,n]$. Suppose $$n=a_1+\ldots+a_r$$ Define a partition of $[1,n]$ as follows:

1. $X_i=[1,a_1]$, if $i=1$;
2. $X_i=[\sum_{j=1}^{i-1} a_j+1,\sum_{j=1}^{i} a_j]$, $2\leq i\leq r$.

Let $S$ be the set of $\tau\in\mathfrak{S}_{n}$ such that $\tau|_{X_i}$ is increasing for $i\in[1,r]$.

If $(Y_1,\ldots,Y_r)$ is a partition of $[1,n]$ such that $|Y_i|=a_i$, for $i\in[1,r]$, then there exists one and only one $\tau\in S$ which maps $(X_1,\ldots,X_r)$ to $(Y_1,\ldots,Y_r)$.

Why do we need $S$ here? e.g. why not just work with $\mathfrak{S}_{n}$, instead? Is $S$ required in order to guarantee the uniqueness of $\tau$?