I am working with a function of type
F[a,b,c,d,e,f]
that obeys the following symmetries:
F[a,b,c,d,e,f]=F[c,d,a,b,e,f]=F[c,d,e,f,a,b],
and
F[a,b,c,d,e,f]=F[b,a,c,d,e,f]=F[a,b,d,c,e,f]=F[a,b,d,c,f,e],
i.e. is symmetric in entries 1-2,3-4 and 5-6 and also is symmetric if I switch any couple of entries with a different one. I want to compute a list with all the possible distinct values of the function, giving different entries. To do this, I want to compute a list of lists containing all the possible distinct orderings of my entries (given the symmetries of my function). From a simple combinatorics argument, I expect
6!/2!2!2!3!=15
different orderings of my entries. To be more specific, the different orderings I need are
{1,2,3,4,5,6},{1,2,3,5,4,6},{1,2,3,6,4,5},{1,3,2,4,5,6},{1,3,2,5,4,6},{1,3,2,6,4,5,},{1,4,2,3,5,6},{1,4,2,5,3,6},{1,4,2,6,3,5},{1,5,2,3,4,6},{1,5,2,4,3,6},{1,5,2,6,3,4},{1,6,2,3,4,5},{1,6,2,4,3,5},{1,6,2,5,3,4}
How can I compute those distinct orderings in Mathematica? Can I exploit some specific features of Permutation[] command?
DeleteDuplicates[f @@@Permutations[{a, b, c, d, e, f}]]
gives all the different results. $\endgroup$