Chop vectors and the lattice of integer partitions

Abstract

The set S consists of all finite sets of integer length sticks. By listing the lengths of these sticks in nonincreasing order, we can represent each element S of S by a nonincreasing sequence of positive integers. These sequences can then be partially ordered by dominance to obtain a lattice (also denoted by S ) closely related to the lattice of integer partitions. The chop vector of an element S ∈ S is defined to be the infinite vector v S = ( v 1 , v 2 , v 3 , … ) , where each v w is the minimum number of cuts needed to chop S into unit pieces, given a knife which can cut up to w sticks at a time. The chop vectors are ordered componentwise. In this paper, we show that the mapping that takes any element of S to its chop vector is order-preserving.