Problem Statement

Niwango bought a piece of land that can be represented as a half-open interval [0, X).

Niwango will lay out N vinyl sheets on this land. The sheets are numbered 1,2, \ldots, N, and they are distinguishable. For Sheet i, he can choose an integer j such that 0 \leq j \leq X - L_i and cover [j, j + L_i) with this sheet.

Find the number of ways to cover the land with the sheets such that no point in [0, X) remains uncovered, modulo (10^9+7). We consider two ways to cover the land different if and only if there is an integer i (1 \leq i \leq N) such that the region covered by Sheet i is different.

Constraints

• 1 \leq N \leq 100
• 1 \leq L_i \leq X \leq 500
• All values in input are integers.

Sample Input 1

3 3
1 1 2

Sample Output 1

10

• If we ignore whether the whole interval is covered, there are 18 ways to lay out the sheets.
• Among them, there are 4 ways that leave [0, 1) uncovered, and 4 ways that leave [2, 3) uncovered.
• Each of the other ways covers the whole interval [0,3), so the answer is 10.

Sample Input 2

18 477
324 31 27 227 9 21 41 29 50 34 2 362 92 11 13 17 183 119

Sample Output 2

134796357

• Find the number of ways modulo (10^9+7).