On Minimal Strings Containing the Elements of Sn by Decimation

One of the most fundamentals objects of combinatorics is perhaps Sn – the set of all permutations of an alphabet of size n. This paper investigates a new problem concerning Sn the strings containing Sn by decimation. Decimation means that each permutation is formed by deleting entries in the string. This investigation has focused on strings of minimal length satisfying the decimation constraint. These strings are refered to these as being optimal solutions. Optimal solutions are not necessarily unique – for example, the reversal of a optimal solution is also an optimal solution. This problem is applicable to a computer graphics rendering. Abstract formulations will be presented, followed by some simple algorithms for generating non-optimal solutions. It will be shown that the optimal solution length is n2 2n 4. Techniques for obtaining optimal solutions are investigated, including exhaustive trial-and-error and Genetic Algorithm (GA) evolution of solution strings. Variations such as alternating groups, and r-permutations of n are also discussed. Applications and related problems such as de Bruijn sequences are presented, followed by the results of computer searches.