of the table is reverse colex order by , which is the same as colex order by . Lex order by is the same as lex order by . Weak order of permutations The set of permutations on n items can be given the structure of a partial order, called the weak order of permutations, which forms a lattice. The Hasse diagram of the inversion sets ordered by the subset relation forms the skeleton of a permutohedron. If a permutation is assigned to each inversion set using the place-based definition, the resulting order of permutations is that of the permutohedron, where an edge corresponds to the swapping of two elements with consecutive values. This is the weak order of permutations. The identity is its minimum, and the permutation formed by reversing the identity is its maximum. If a permutation were assigned to each inversion set using the element-based definition, the resulting order of permutations would be that of a Cayley graph, where an edge corresponds to the swapping of two elements on consecutive places. This Cayley graph of the symmetric group is similar to its permutohedron, but with each permutation replaced by its inverse. See also Factorial number system Permutation graph Transpositions, simple transpositions, inversions and sorting Damerau–Levenshtein distance Parity of a permutation Sequences in the OEIS: Sequences related to factorial base representation Factorial numbers: and Inversion numbers: Inversion sets of finite permutations interpreted as binary numbers: (related permutation: ) Finite permutations that have only 0s and 1s in their inversion vectors: (their inversion sets: ) Number of permutations of n elements with k inversions; Mahonian numbers: (their row maxima; Kendall-Mann numbers: ) Number of connected labeled graphs with n edges and n nodes: References Source bibliography Further reading Presortedness measures Permutations Order theory String metrics Sorting algorithms Combinatorics Discrete mathematics