Chop vectors and the lattice of integer partitions

The set SS consists of all finite sets of integer length sticks. By listing the lengths of these sticks in nonincreasing order, we can represent each element SS of SS by a nonincreasing sequence of positive integers. These sequences can then be partially ordered by dominance to obtain a lattice (also denoted by SS) closely related to the lattice of integer partitions. The chop vector   of an element S∈SS∈S is defined to be the infinite vector vS=(v1,v2,v3,…), where each vwvw is the minimum number of cuts needed to chop SS into unit pieces, given a knife which can cut up to ww sticks at a time. The chop vectors are ordered componentwise. In this paper, we show that the mapping that takes any element of SS to its chop vector is order-preserving.