Hidden Pancakes timer 30s memory 1024MB

We are cooking \mathbf{N} pancakes in total. We cook one pancake with a 1 centimeter (cm) radius, one with a 2 cm radius, one with a 3 cm radius, ..., and one with an \mathbf{N} cm radius, not necessarily in that order. After we cook the first pancake, we just lay it on a plate. After we cook each subsequent pancake, we lay it on top of the previously made pancake, with their centers coinciding. In this way, a pancake is visible from the top of the stack when we first add it. A pancake only becomes hidden if we later cook another pancake with a larger radius.

For example, say we cook 4 pancakes. We first cook the pancake with radius 3 cm, and it is visible. Then, we cook the pancake with radius 1 cm, lay it on top of the first one and both are visible. Third, we cook the pancake with radius 2 cm, and now that covers the previous pancake, but not the first one, so 2 pancakes remain visible in total. Finally, we cook the pancake with radius 4 cm which covers the other pancakes leaving only 1 visible pancake. The picture below illustrates the state of the stack after each pancake is cooked. Within each stack, the fully colored pancakes are visible and the semi-transparent pancakes are not visible.

Let \mathbf{V_i} be the number of visible pancakes when the stack contains exactly i pancakes. In the example above, \mathbf{V_1} = 1, \mathbf{V_2} = 2, \mathbf{V_3} = 2, and \mathbf{V_4} = 1.

Given the list \mathbf{V_1}, \mathbf{V_2}, \dots, \mathbf{V_N}, how many of the \mathbf{N}! possible cooking orders yield those values? Since the output can be a really big number, we only ask you to output the remainder of dividing the result by the prime 10^9+7 (1000000007).