movement rule: right-else-left-else-stop
               =========================

note that the starting position of this robot is sideways in the dead
end, otherwise it can't get out.
with (i,j)||j meaning the robot is in (i,j) parallel to the j-axis,
the only possible single-step movements in this scenario are
   from (i,j)||j to (i+/-1,j)||i and
   from (i,j)||i to (i,j+/-1)||j.
so, the only possible multi-step movements are
   from (i,j)||j to (i+2n-1,j+2m)||i or to (i+2n,j+2m)||j; and
   from (i,j)||i to (i+2n-1,j+2m)||j or to (i+2n,j+2m)||i.
from this we see that the robot can't return to the dead end where it
started from, because if it's aligned parallel to the j-axis at the
start, it would have to finish in the same dead end parallel to the
i-axis (otherwise it can't get in) which is impossible according to
the above movement possiblities.
so, the permutations (1)(...) cannot be realized.

now suppose the robot finished in some dead end at (i,j)||i, then in
order to start its next trip, it is rotated 90 degrees to (i,j)||j so
it can leave from the dead end. so, we get the following possible
movements from a position in trip k to a position in trip k+1:
   from (i,j)||j to (i+2n-1,j+2m)||j or to (i+2n,j+2m)||i; and
   from (i,j)||i to (i+2n-1,j+2m)||i or to (i+2n,j+2m)||j.
now the robot might be able to finish in the dead end where it started
trip k. extending this, we see that permutations (123..2n+1)(...)
cannot be realized, but that permutations (123..2n)(...) might be
realized.


permutation      no indoor walls           indoor walls allowed
-----------      ---------------           --------------------

(1)(...)         not possible              not possible
(123)(...)
(12345)(...)
etc.
                 1_ _2
(12)             |_ _|                     same

                    _4
(1234)           1_| |_                    same
                 |_   _|
                   |_| 3
                   2

                    _1
(12)(34)         2_| |_ _4                 same
                 |_ _   _|
                     |_|
                     3
                                             6_ _5
                                            _| | |_
(123456)         not possible              |_     _|
                                           1 |_|_| 4
                                             2   3

                          _5                  _1
(12)(3456)              _| |_              2_| |_ _
                      _|  _  |_            |_ _    |_
                    _|  _|_|_  |_            |_     _|
                 6_|  _|1| |_|_  |_          3 |_|_| 6
                 |_  |_2>|  _|_|  _|           4   5
                   |_  |_| |_|  _| 4
                     |_       _|
                       |_   _|
                         |_|
                         3

                   1_   _4
(12)(34)(56)      _| |_| |_ _6             same
                 |_ _   _   _|
                 2   |_| |_|
                     3     5