We prove that if σ is a permutation of Sn with m cycles of lengths l1, . . . , lm, the subset of the permutahedron Πn fixed by the natural action of σ is a polytope with volume n m−2 gcd(l1, . . . , lm).

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Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections to combinatorics, both in the sense of enumeration (e.g. Ehrhart theory) and combinatorial structures (e.g. graphs and permutations).Definition 2.1. Consider the polytopes, P 1 , P 2 , . . . , P m ⊂ R n . We define the Minkowski sum of the m polytopes as P 1 + P 2 + · · · + P m := {x 1 + x 2 + · · · + x m : x j ∈ P j for 1 ≤ j ≤ m} .Given v, w ∈ R n , we write [v, w] for the line segment from v to w.