Let $n\in\mathbb{N}$, and let $\sigma\in S_n$ be a permutation in the sense of \ref{def:permutation-initial-segment-2026a}. For each $r\in\{1,\dots,n-1\}$, define the adjacent transposition $\tau_r\in S_n$ by

and $\tau_r(m)=m$ for every $m\in\{1,\dots,n\}\setminus\{r,r+1\}$. Then there exist an integer $N\ge 0$ and indices $r_1,\dots,r_N\in\{1,\dots,n-1\}$ such that