9.3 The Macdonald Group $G(2,8)$ with different orderings

This behaviour is rather discouraging as good combinations for small $m$ become bad ones for large $m$ and vice versa. It looks like, though, that from $m = 8$ on, the behaviour gets stable. Thus choosing $m = 8$ as starting point for larger examples seems to be a good idea. Therefore $G(2,8)$ was computed using the strategies NONE, P-R, I-ALL, and I-R combined with 625 kbos and all length-lexicographic (24) and syllable-left (24 as well) orderings. Thus a total of 2792 combinations was computed and evaluated.

For the length-lexicographical orderings all strategies behave pretty uniform. Strategy NONE enumerates 1343/1343 cosets maximal/totally for all, I-R between 1718/1719 and 1780/1780, P-R between 5261/5261 and 5340/5340, and I-ALL between 6725/6725 and 6869/6869 cosets. Notice, that the maximal number of cosets defined is equal to the total number of cosets defined for all combinations, except some combinations with I-R were we have defined one additional coset.

For the kbos there is a greater variation of the results. For the strategy NONE we have that between 1210/1210 and 1343/1343 cosets were enumerated maximal/totally. Best performed those with the following weights: $(a\; 5), (b\; 1), (A\; 5), (B\; 1)$, worst those with equal weights for all letters which is in fact a length-lexicographical ordering which is used as tie breaker for kbos. That is, all kbos performed equally or better than the length-lexicographical ordering. The strategy I-R showed a greater variation. Between 1032/1033 and 2357/2359 cosets were enumerated maximal/totally. Best were those kbos attaching a high weight to $b$ and low weights to the other letters of the alphabet. The best combination of weights was $(a\; 1), (b\; 5), (A\; 1), (B \;1)$ in contrast to strategy NONE. The worst combinations have high weights attached to $A$ and $B$ and low weights to $a$, and $b$. Strategy I-ALL yielded the following results: between 2067/2067 and 7235/7235 cosets were enumerated maximal/totally. This strategy was better than I-R for some kbos. It also had the largest variation. Good kbos were those with a high weight attached to $B$ and low weights to the other letters of the alphabet. Worst combinations were those attaching high weights to $a$ and $A$ and low weights to $b$ and $B$. Finally, strategy P-R enumerated between 3776/3776 and 6948/6948 cosets maximal/totally together with kbos. It has a smaller variation than I-ALL. Good kbos were those with a high weight attached to $B$ and low weights to the other letters of the alphabet as for I-ALL. But in contrast bad kbos were those attaching high weights to $a$ and $b$ and low weights to $A$ and $B$, just the opposite to strategy NONE.

Finally for the syllable orderings we got the following results. Contrary to the length lexicographical orderings which did not differ very much from each other the syllable orderings show great veriations. For NONE we have that the best ordering is syl-l-AbBa with 1817/1845 cosets defined maximal/totally, while syl-l-abBA is worst with 12748/14743 cosets defined. For I-R the best ordering is syl-l-bBAa with 1453/1660 cosets defined, while syl-l-AaBb is worst with 13272/14788 cosets defined. For I-ALL the best ordering is syl-l-baAB with 1700/1862 cosets defined, while syl-l-bAaB is worst with 18447/20704 cosest defined and syl-l-bBAa was not computed at all. Finally for P-R the best ordering is syl-l-BbaA with 2389/2399 cosets defined, while syl-l-AbBa is worst with 4847/4849. Strategy P-R showed the slightest variation while for the other strategies the worst syllable orderings are up to 10 times worse than the best one. Further, non syllable ordering is very good or very bad for alll strategies. It should be noted, too, that the best ordering for I-R is second best for NONE and vice versa.

To sum up the results, we have that the Knuth-Bendix orderings are best for strategy NONE followed by the length-lexicographical orderings. All syllable orderings are considerably worse for this strategy. Strategy I-R shows a similar behaviour, but this time the length-lexicographical orderings enumerate as much cosets as the Knuth-Bendix orderings on average, that is about half of them enumerate fewer cosets while the other half enumerates more cosets. Most syllable orderings enumerate more cosets than the Knuth-Bendix orderings. As for strategy I-ALL, we have that almost all Knuth-Bendix orderings perform better than the length-lexicographical orderings. Nevertheless, the best and worst orderings are a syllable orderings. Finally, for strategy P-R we have similar to I-R that the length-lexicographical orderings enumerate as much cosets as the Knuth-Bendix orderings on average, but other Knuth-Bendix orderings are best. All syllable orderings are better than the length-lexicographical ordering and most better than the Knuth-Bendix orderings.

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