Inversion (discrete mathematics): Difference between revisions

Revision as of 16:21, 1 October 2022

Revision as of 21:57, 1 October 2022

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[[File:2-element subsets of 4 elements; array, hexagonal.svg|thumb|The six possible inversions of a 4-element permutation]]

The following sortable table shows the 24 permutations of four elements (in the <math>\pi</math> column) with their place-based inversion sets (in the p-b column), inversion related vectors (in the <math>v</math>, <math>l</math>, and <math>r</math> columns), and inversion numbers (in the # column). (The small columns with smaller print and no heading are reflections of the columns next to them, and can be used to sort them in [[colexicographic order]].)

It can be seen that <math>v</math> and <math>l</math> always have the same digits, and that <math>l</math> and <math>r</math> are both related to the place-based inversion set. The nontrivial elements of <math>l</math> are the sums of the descending diagonals of the shown triangle, and those of <math>r</math> are the sums of the ascending diagonals. (Pairs in descending diagonals have the right components 2, 3, 4 in common, while pairs in ascending diagonals have the left components 1, 2, 3 in common.)