Problem:
Score : 1100 points

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.

Input

Input is given from Standard Input in the following format:

N X
L_1 L_2 \ldots L_N

Output

Print the answer.

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).

Solution in PyPy3 (2.4.0) (judge verdict: Runtime Error)

N,X=map(int,input().split())
L=sorted(map(int,input().split()),reverse=True)
mod=10**9+7

D=[[0]*(X+1) for i in range(N//2+1)]
D[1][L[0]]=1

for l in L[1:]:
    ND=[[0]*(X+1) for i in range(N//2+1)]

    for c in range(1,N//2+1):
        for s in range(X+1):
            if D[c][s]==0:
                continue
            
            k=D[c][s]

            if s+l+c<=X and c+1<=N//2:
                ND[c+1][s+l]=(ND[c+1][s+l]+k*(c+1))%mod
            ND[c][s]=(ND[c][s]+k*(s-c*(l-1)))%mod

            for i in range(1,min(X-s-c+2,l+1)):
                ND[c][s+i]=(ND[c][s+i]+k*2*c)%mod

            if c==1:
                continue

            for i in range(1,min(X-s-c+3,l+1)):
                ND[c-1][s+i]=(ND[c-1][s+i]+k*(c-1)*(l-i+1))%mod
    D=ND

print(D[1][X])