Greetings to all.
As some of you may remember I made several personal story type posts concerning my progress in solving enumeration problems with the Polya Enumeration Theorem (PET). This theorem would seem to be among the most exciting in mathematics and it is of an amazing simplicity so that I never cease to suggest to mathematics teachers to present it to gifted students even before university. My previous efforts are documented at your site, namely at this MaplePrimes link I and this MaplePrimes link II.
I have been able to do another wonderful cycle index computation using Maple recently and I would like to share the Maple code for this problem, which is posted at Math StackExchange.com (this post includes the code) This time we are trying to compute the cycle index of the automorphism group of the 3-by-3-by-3 cube under rotations and reflections. I suggest you try this problem yourself before you look at my solution. Enjoy!
I mentioned in some of my other posts concerning PET that Maple really ought to come with a library of cycle indices and the functions to manipulate them. I hope progress has been made on this issue. I had positive feedback on this at the time here at your website. Do observe that you have an opportinuity here to do very attractive mathematics if you prepare a worksheet documenting cycle index facilities that you may eventually provide. This is good publicity owing to the fact that you can include images of the many geometric objects that appear which all look quite enticing and moreover potential readers get rewarded quickly as they discover that it takes little effort to master this theorem and proceed to work with symmetries themselves and investigate them. This sort of thing also makes nice slides.
With best wishes for happy combinatorics computing,