2048 is played in a 4*4 space. At the start there are 2 tiles spawned.
Move - a swipe followed by a new tile (either 2 or 4) spawning.
Position - any configuration of the various tiles and blanks on the board.
End Position - a position where no moves are possible (must have either 2 or 4 on one of outer 12 blocks to be obtainable)
Valid Position - obtainable through play.
Invalid Position - unobtainable through play. (all 2's)
For a given position p and natural number n, let $n(p)$ be the number of possible positions n moves earlier. Let $n^+(p)$ be the sum of all the different ways of each position from n(p) to reach position p.
Note that $n^+(p) \geq n(p)$; there are two ways figure 1 could transpose to figure 2.
figure1______________________________figure2
Example where $1(p) = 1^+(p)$: 
Is it possible to backtrack from a valid position and get an invalid position? If so, can n(p) not increase when add 1 to n?
How to tell valid positions from invalid?