Think of permutations as acyclic orientations on a complete graph. One reaches neighboring permutations by edge flips that preserve acyclicity; these correspond to adjacent pair swaps. Each edge acts as a toggle switch, so every cycle in the set of permutations is even, and the parity of a permutation is well-defined.

The illustration is a permutahedron as zonotope (Mercator projection). Classification of Vertex-Transitive Zonotopes gives a survey and generalization of this point of view.

Think of permutations as acyclic orientations on a complete graph. One reaches neighboring permutations by edge flips that preserve acyclicity; these correspond to adjacent pair swaps. Each edge acts as a toggle switch, so every cycle in the neighbor graph is even, and the parity of a permutation is well-defined.

The illustration is a permutahedron as zonotope (Mercator projection). Classification of Vertex-Transitive Zonotopes gives a survey and generalization of this point of view.

Think of permutations as acyclic orientations on a complete graph. One reaches neighboring permutations by flipping edges. Each edge acts as a toggle switch, so every cycle in the set of permutations is even, and the parity of a permutation is well-defined.

Think of permutations as acyclic orientations on a complete graph. One reaches neighboring permutations by edge flips that preserve acyclicity; these correspond to adjacent pair swaps. Each edge acts as a toggle switch, so every cycle in the set of permutations is even, and the parity of a permutation is well-defined.

The illustration is a permutahedron as zonotope (Mercator projection). Classification of Vertex-Transitive Zonotopes gives a survey and generalization of this point of view.