1. a[0] = 0, a[1] = 1 if N > 1
2. For n > 0, a[n] <= a[n - 1] + 1
Let's start with DP. Let D[x][y] be the number of arrays a[0 ... x] with a[x] >= y. Then, the number of arrays a[0 ... x] such that a[x] == y is exactly D[x - 1][y - 1] because a[x - 1] must be no less than y - 1. Thus we have relation
D[x][y] = D[x - 1][y - 1] + D[x][y + 1] (*)
with some boundary conditions, including D[x][0] = D[x][1]. For simplicity we can append a[N] = 1 in order to find the final answer. Suppose a[x0] and a[x1] are the only fixed numbers in [x0, x1], the core task is to compute f(x0, y0, x1 - 1, y1 - 1) = D[x1 - 1][a[x1] - 1] from D[x0][0] = D[x0][1] = ... = D[x0][a[x0]] = 1, and the final answer will be the product of all such f(x0, y0, x1, y1) modulo M.
With initial observations, I almost felt f(x0, y0, x1, y1) has binomial form C(a, b), although it doesn't because of boundary condition D[x][0] = D[x][1]. After some calculations, I noticed the nice relation
f(x0, y0, x1, y1) = C(2W - H, W) - C(2W - H, W - y1 - 1)
where W = x1 - x0, H = y1 - y0. The proof is actually mathematically very interesting.
Imagine you're on an infinite xy-plane and stand at cell (0, y0). Relation (*) says you can only go in two directions (1, 1) or (0, -1). Denote them by step A and B. Then D[x1][y1] is the number of shortest path you may take to get to cell (x1 - x0, y0), and the constraint D[x][0] = D[x][1] implies you are not allowed to go below x-axis. Now, C(2W - H, W) is the number of shortest paths you could take without this contraint. This is because to go from (0, y0) to (x1 - x0, y1), the number of A step you need is exactly x1 - x0 = W since only A step makes x-direction move. The number of required B step to count for y-direction move is hence y0 - y1 + W = W - H. We could also transform the xy-plane to x'y'-plane with x' = x and y' = y - x. There A step become horizontal step (1, 0) and B step is vertical downward (0, -1), i.e. everything is square and not tilted.
We now have to show C(2W - H, W - y1 - 1) is number of shortest paths that goes under x-axis in xy-plane. Going under x-axis is the same as stepping into zone x' + y' < 0 since y = x + y' = x' + y'. What is the number of such shortest paths? For illustration we draw an example of y0 = 2, W - H = 6, x1 - x0 = 13 in x'y'-plane below, where the goal is to go from upper left corner to lower right corner by stepping in at least one '*' cell, i.e. all '*' cells have x' + y' < 0.
(0, 2)
oooooooooooooo
oooooooooooooo
(0, 0)oooooooooooooo
*ooooooooooooo
**oooooooooooo
***ooooooooooo
****oooooooooo (13, -4)
Here's the magic: we can 'fold' the plane along x' + y' = 0 to get this:
(0, 2)
oooooooooooooo
oooooooooooooo
(0, 0)oooooooooooooo
*ooo----------
**oo----------
***o----------
****---------- (13, -4)
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++ (3, -14)
Fold all the '-' cells to the '+' cells shown above. There's then a 1-1 mapping between the original shortest paths from (0, 2) to (13, -4) touching at least one '*' cell and the shortest paths from (0, 2) to (3, -14), and so the number of shortest paths is C(19, 3). To generalize, with some counting we can thus prove that it's C(2W - H, W - H + 1 - (y0 + 1) - 1) = C(2W - H, W - y1 - 1)
Overall, this solution takes O(N) computation of binomial coefficient C(a, b). Assuming each C(a, b) requires O(b)log(M) where M is 10^9+7 and b = O(W), each boils down to O(W)log(M). The total time complexity is then O(N)log(M) and space complexity O(1).
