Adjacent Vertices on a Permutohedron

Abstract

The convex hull $P_n $ of \[S = \left\{ {\left( {a_{\pi ( 1 ),} a_{\pi ( 2 )} , \cdots ,a_{\pi ( n )} } \right)| {\pi {\text{ is a permutation of }}( {1,2,3, \cdots ,n} )} } \right\} ,\] where $a_1 ,a_2 , \cdots ,a_n $ are integers, is defined as a permutohedron. If $a_1 < a_2 < \cdots < a_n $, it is shown that every vertex of $P_n $ has $n - 1$ adjacent vertices and a method for determining the adjacent vertices is given. The algebraic description of $P_n $ is given by considering its facets.