Greedy Linear Extensions For Minimizing Bumps

Abstract

Let P be a finite poset and let L={x 1<...<Xn} be a linear extension of P. A bump in L is an ordered pair (x i , x i+1) where x i<Xi+1 in P. The bump number of P is the least integer b(P), such that there exists a linear extension of P with b(P) bumps. We call L optimal if the number of bumps of L is b(P). We call L greedy if x i <X< i>j for every j>i, whenever (x i, x i+1) is a bump. A poset P is called greedy if every greedy linear extension of P is optimal. Our main result is that in a greedy poset every optimal linear extension is greedy. As a consequence, we prove that every greedy poset of bump number k is the linear sum of k+1 greedy posets, each of bump number zero.