\notag\end{aligned}$$

\[lem2..5\] Let $\sigma\in S_{2d}$, $\sigma'\in S_{2d'}$, and $\sigma''\in S_{2d''}$ be such that $d'+d''=d$ and $$\sigma(r):=\begin{cases}
\sigma'(r)&\text{ if }1\leq r\leq 2d',\\
\sigma''(r-2d')+2d'&\text{ if }2d'+1\leq r\leq 2d.
\end{cases}$$ Then $\bft_\sigma=
\bft_{\sigma'}\bft_{\sigma''}$ as elements of the superalgebra $\sP(W)\otimes \sS(W)$.

Follows immediately from the explicit summation formula for $\bft_\sigma$ given in .

\[taud0dr\] Let $\sigma\in S_{2d}$. Then the following statements hold.

-   $\bft_\sigma=\bft_{\sigma'\sigma\sigma''}$ for every $\sigma',\sigma''\in H_{2d}$.

-   There exist $\sigma',\sigma''\in H_{2d}$ and integers $0=d_0<d_1< \cdots< d_{r-1}<d_r=d
    $ such that $$\label{eqTAUdecc}
    \sigma'\sigma\sigma''=\tau_{d_0,d_1}^{}
    \tau_{d_1,d_2}^{}\cdots \tau_{d_{r-1},d_r^{}}^{},$$ where $\tau_{a,b}$ for $a<b$ denotes the cycle $
    (2a+2,2a+4,\ldots,2b-2,2b)
    $.

\(i) Follows from .

\(ii) First we show that there exist integers $a_1,\ldots,a_d,b_1,\ldots,b_d$ and a permutation $\tau\in S_d$ such that $$2s-1\leq a_s,b_s\leq 2s
\text{ and }
\sigma(a_s)=b_{\tau(s)}
\text{ for }
1\leq s\leq d.$$ To this end, we construct an undirected bipartite multigraph $\mathcal G$, with vertex set $$\mathcal V
:=\{\mathbf a_1,\ldots,\mathbf a_d\}\cup\{\mathbf b_1,\ldots,\mathbf b_d\},$$ and $m_{s,s'}:=\left|M_{s,s'}\right|$ edges between $\mathbf a_s$ and $\mathbf b_{s'}$, labelled by elements of the set $M_{s,s'}$, where $$M_{s,s'}:=\{\sigma(2s-1),\sigma(2s)\}\cap \{2s'-1,2s'\}\,
\text{ for all }1\leq s,s'\leq d
.$$ Elements of $M_{s,s'}$ correspond to equalities of the form $\sigma(a_s)=b_{s'}$, where $2s-1\leq a_s\leq 2s$ and $2s'-1\leq b_{s'}\leq 2s'$. Every vertex in $\mathcal G$ has degree two, and therefore $\mathcal 