Peter Ungar

2N Noncollinear points determine at least 2N directions

Abstract We find sharp bounds for the number of moves required to bring a permutation to the form n ( n −1),…, 1 if a move consists of inverting some increasing substrings. If we invert every maximal increasing substring in each move we need at most n − 1 moves. If n is even and we start with 1, 2,…, n and we do not invert the entire permutation at once, then we need at least n moves. The lower bound implies that when n ⩾ 4 is even, n points which are not collinear determine at least n different directions, as do n + 1. These bounds are sharp.