In this chapter there are two projects. The first one (A) is about random permutations of a finite set, cycles and permutation matrices. The second (B) is an investigation of card shuffling, introducing many of the standard ideas including perfect and approximate riffle shuffles.

A Cycle decompositions

Aims of the project

We shall use MATLAB to investigate ‘random’ permutations, especially their disjoint cycle decompositions. There is a theoretical and experimental investigation of the [5 [5 average number of disjoint cycles occurring in a random permutation. The basic material on permutations is generally covered in a first course on abstract algebra; see, for example, [11].

Mathematical ideas used

This investigation studies permutations of a finite set, decompositions into disjoint cycles, the order of a permutation and permutation matrices. The order of a permutation involves the idea of the least common multiple (lcm) of a set of integers. Also the average number of disjoint cycles occurring in permutations of a given finite set is investigated both experimentally and theoretically. Note: We always write composition of permutations from right to left: the notation Φ2Φ1 means ‘do Φ1 first and then do Φ2’.

MATLAB techniques used

A given M-file produces ‘random’ permutations of the consecutive numbers 1,2,…, n, and another breaks a permutation up into disjoint cycles.