Exercise
For each pair \( k, \ell \in \mathbb{N}_{n} \) with \( k<\ell \), prove that:
\( \Delta \pi^{k} \nearrow \ell(i):=\left\{\begin{array}{ll}i & \text { if } i<k \text { or } i>\ell \text {; } \\ i-1 & \text { if } k<i \leq \ell ; \\ \ell & \text { if } i=k .\end{array}\right. \)
\( \triangle \pi^{k \nearrow k}=\pi^{k} \searrow k=\iota \)
\( \rightarrow \pi^{k} \nearrow \ell \) and \( \pi^{\ell}{ }^{\lambda} \) are inverses
- \( \pi^{k \nearrow \ell}=\pi^{1,2, \ldots, k-1, k+1, \ldots, \ell-1, \ell, k, \ell+1, \ldots, n} \)
- \( \pi^{\ell} \searrow k=\pi^{1,2, \ldots, k-1, \ell, k, k+1, \ldots, \ell-2, \ell-1, \ell+1, \ldots, n} \)
1. Note that \( \pi^{k} \nearrow \ell \) moves \( k \) up to the \( \ell \) th position,
