SIMON-MARAIS 2019 A2

Consider the operation $\ast$ that takes pair of integers and returns an integer according to the rule $$a\ast b=a\times (b+1).$$ [list=a] []For each positive integer $n$ , determine all permutations $a_1,a_2,\dotsc , a_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$(\cdots ((a_1\ast a_2)\ast a_3) \ast \cdots \ast a_{n-1})\ast a_n.$$ [/] []For each positive integer $n$ , determine all permutations $b_1,b_2,\dotsc , b_n$ of the set $\{ 1,2,\dotsc ,n\}$ that maximise the value of $$b_1\ast (b_2\ast (b_3\ast \cdots \ast (b_{n-1}\ast b_n)\cdots )).$$ [/] [/list]