1.1 Descents of a permutation
Let \(\sigma \in \mathfrak {S}_{n+1}\). Position \(i \in \{ 0, \dots , n-1\} \) is a descent of \(\sigma \) when \(\sigma (i+1) {\lt} \sigma (i)\). The descent set is
\[ D(\sigma ) = \{ i : \sigma (i+1) {\lt} \sigma (i) \} \subseteq \{ 0, \dots , n-1\} . \]
Set \(\operatorname {des}(\sigma ) = |D(\sigma )|\) and \(\operatorname {asc}(\sigma ) = n - \operatorname {des}(\sigma )\).
\(\operatorname {des}(\mathrm{id}_{\mathfrak {S}_{n+1}}) = 0\).
Let \(\rev _n \in \mathfrak {S}_{n+1}\) be the order-reversing permutation \(i \mapsto n - i\). Position \(i\) is a descent of \(\sigma \) if and only if \(i\) is not a descent of \(\rev _n \cdot \sigma \), hence \(\operatorname {des}(\rev _n \cdot \sigma ) = n - \operatorname {des}(\sigma )\). In particular \(\operatorname {des}(\rev _n) = n\).