belongs to a non-trivial orbit of $\mathfrak{G}$, $u_i$ belongs to some non-trivial orbit of $\mathfrak{G}_{u_1,\ldots,u_{i-1}}$ ($i=2,\ldots,s$) and $\mathfrak{G}_{u_1,\ldots,u_{s}} = \{ 1 \}$, where $\mathfrak{G}_{u_1,\ldots,u_{i-1}} = \left\{ \gamma \in \mathfrak{G} : \gamma \hspace{0.5mm} u_k = u_k, k = 1,\ldots,i-1 \right\}$. Let $x_1,x_2,\ldots,x_{s}$ and $y_1,y_2,\ldots,y_{s}$ be two fastening sequences of $\mathfrak{G}$. A moment’s reflection would show that in order to determine a permutation in $\mathfrak{G}$ mapping $x_1$ to $y_1$, we only need to work out two group of partitions $\Pi^*_{x_1},\Pi^*_{x_1,x_2},\ldots,\Pi^*_{x_1,\ldots,x_{s}}$ and $\Pi^*_{y_1},\Pi^*_{y_1,y_2},\ldots,\Pi^*_{y_1,\ldots,y_{s}}$ and to know the corresponding relation between cells of partitions in each pair $( \Pi^*_{x_1,\ldots,x_k},\Pi^*_{y_1,\ldots,y_k} )$, $k=1,\ldots,s$.

In the 3rd section, we will show how to work out those partitions and determine the corresponding relation between cells of partitions in each pair. For convenience, we use the term “[*information about $\mathfrak{G}$*]{}” to describe the information about the partition $\Pi_G^*$ and a series of partitions of $[n]$ associated with a fastening sequence of $\mathfrak{G}$.

The Partition $\Pi[ \oplus V_{\lambda};v ]$ — an Approximation to $\Pi^*_v$
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Obviously, all permutations of $[n]$ form a group under composition of maps, which is called the [*symmetric group of degree*]{} $n$ and denoted by ${\mathrm{Sym} \hspace{0.4mm} [n]}$, or by $S_n$ for short. Each permutation $\sigma$ 