Let $\mathrm{rank}_I$ be the rank relative to the insertion order for $\sigma \in S_9$ . If $\mathrm{rank}_I(\sigma) = 9 \times 8n+23$, for some $n \in \mathbb{N}$, which value will assume $\sigma(7)$ ?
Each tree's node of the form $a_1a_2a_3a_4a_5a_6a_7$ has $8$ children and these children have $9$ children by themselves. Starting at one of these nodes, I can visit the children of the first child of the node I chose and then the children of the subsequent one and finally after visiting $5$ children of the third one, the next child will be the permutation that satisfies the above condition. By this method, $\sigma(7)$ seems to be equal to $8$. Should I rely on this method?