Question 43
A school has $2020$ pupils, no two of whom are the same height. It also has a long bench on which the pupils can all sit in some order. An *arrangement* means some ordering of pupils on the bench. A *swap* means two pupils exchanging places while the other $2018$ stay fixed. An arrangement is said to be *unimodal* if the heights increase in sequence up to the tallest member and then decrease in sequence. For this purpose, the increasing arrangement and the decreasing arrangement are also considered unimodal.
The pupils start the day sorted from shortest to tallest. They want to re-arrange themselves to the opposite order, from tallest to shortest, in a sequence of swaps, such that all intermediate arrangements are unimodal.
Can this be done?
The pupils start the day sorted from shortest to tallest. They want to re-arrange themselves to the opposite order, from tallest to shortest, in a sequence of swaps, such that all intermediate arrangements are unimodal.
Can this be done?