This stuff is inspired by the following article:
J. H. Conway, N. J. A. Sloane and A. R. Wilks, Gray Codes for Reflection Groups, Graphs and Combinatorics, 5 (1989), pp. 315-325.
(See item 152 at http://www.research.att.com/~njas/doc/pub.html
or directly at http://www.research.att.com/~njas/doc/wilks.ps
and the assorted figures Fig. 1 , Fig. 2 , Fig. 3 ,
Fig. 4 and Fig. 5 in separate files.)
and by
F. Ruskey and Carla Savage, Hamilton Cycles which Extend Transposition Matchings in Cayley Graphs of Sn,
SIAM Journal on Discrete Mathematics, 6 (1993) 152-166.
(See http://www.csr.uvic.ca/~fruskey/Publications/)
The assorted Maple-procedures can be found near the end of the
document http://www.megabaud.fi/~karttu/matikka/findnext.txt
csw_perm_list_aux(4,[[3,1],[1,3],[2,5]]) -> (cb(ab)2cb)3 = (cb)((ab)2(cb)2)2((ab)2cb)
1: [1 2 3 4]=[] (3 4) = c
2: [1 2 4 3]=((3 4)) (2 3) = b
3: [1 4 2 3]=((2 4 3)) (1 2) = a
4: [4 1 2 3]=((1 4 3 2)) (2 3) = b
5: [4 2 1 3]=((1 4 3)) (1 2) = a
6: [2 4 1 3]=((1 2 4 3)) (2 3) = b
7: [2 1 4 3]=((1 2)(3 4)) (3 4) = c
8: [2 1 3 4]=((1 2)) (2 3) = b
9: [2 3 1 4]=((1 2 3)) (3 4) = c
10: [2 3 4 1]=((1 2 3 4)) (2 3) = b
11: [2 4 3 1]=((1 2 4)) (1 2) = a
12: [4 2 3 1]=((1 4)) (2 3) = b
13: [4 3 2 1]=((1 4)(2 3)) (1 2) = a
14: [3 4 2 1]=((1 3 2 4)) (2 3) = b
15: [3 2 4 1]=((1 3 4)) (3 4) = c
16: [3 2 1 4]=((1 3)) (2 3) = b
17: [3 1 2 4]=((1 3 2)) (3 4) = c
18: [3 1 4 2]=((1 3 4 2)) (2 3) = b
19: [3 4 1 2]=((1 3)(2 4)) (1 2) = a
20: [4 3 1 2]=((1 4 2 3)) (2 3) = b
21: [4 1 3 2]=((1 4 2)) (1 2) = a
22: [1 4 3 2]=((2 4)) (2 3) = b
23: [1 3 4 2]=((2 3 4)) (3 4) = c
24: [1 3 2 4]=((2 3)) (2 3) = b
If we limit ourselves to evenly spaced splicing (each splice-point 6 steps
from its neighbours), there are three alternative ways A,
B and C to continue
this kind of ordering to S5. (Because double cases of ending 4's occur
in above ordering only in three locations, 24-1 (i.e. 0-1), 8-9 and 16-17.