A surprisingly simple algorithm for the generation of all permutations is obtained by mixed radix counting with the radices [2, 3, 4, . . .] (column digits in figure 10.4-A). Whenever the first j digits change with [fxtbook draft of 2008-January-19] 232 Chapter 10: Permutations an increment then the permutation is updated by reversing the first j + 1 elements. As with lex order the first half of the permutations are the complements of the permutations in the second half, now rewrite R = I C I (10.4-1) to see that the first half of the inverse permutations are the reversed inverse permutations in the second half. This can (for n even) also be observed from the positions of the largest element in the inverse permutations.

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Counting sort

The counting sort algorithm sorts an array’s contents by counting the repetition of each element that occurs in the array. A counting sort algorithm is a stable algorithm that works the best when the range of elements in the array is small, say for example from 0 to 9. The count which is the number of repetitions of an element in the array is mapped to an index of an auxiliary array, which is then saved in an auxiliary array. The counting sort algorithm has a stable time complexity of the order of O(size of array + size of auxiliary array). Let’s discuss the algorithm in detail as follows.