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 Rust (1.15.1) (judge verdict: Accepted)

// ---------- begin ModInt ----------
const MOD: u32 = 1_000_000_007;

#[derive(Clone, Copy)]
struct ModInt(u32);

impl std::ops::Add for ModInt {
    type Output = ModInt;
    fn add(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = ModInt;
    fn sub(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + MOD - rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = ModInt;
    fn mul(self, rhs: ModInt) -> Self::Output {
        ModInt((self.0 as u64 * rhs.0 as u64 % MOD as u64) as u32)
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl std::ops::Neg for ModInt {
    type Output = ModInt;
    fn neg(self) -> Self::Output {
        ModInt(if self.0 == 0 {0} else {MOD - self.0})
    }
}

impl std::fmt::Display for ModInt {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::str::FromStr for ModInt {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

#[allow(dead_code)]
impl ModInt {
    pub fn new(n: u32) -> ModInt {
        ModInt(n % MOD)
    }
    pub fn zero() -> ModInt {
        ModInt(0)
    }
    pub fn one() -> ModInt {
        ModInt(1)
    }
    pub fn pow(self, mut n: u32) -> ModInt {
        let mut t = ModInt::one();
        let mut s = self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(self) -> ModInt {
        self.pow(MOD - 2)
    }
}
// ---------- end ModInt ----------

macro_rules! get_mut {
    ($t:tt ; $pos:expr) => {
        $t.get_mut($pos)
    };
    ($t:tt ; $pos:expr, $($rest:expr),*) => {
        $t.get_mut($pos).and_then(|v| get_mut!(v; $($rest),*))
    };
}

use std::io::Read;
use std::cmp::*;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    let n: usize = it.next().unwrap().parse().unwrap();
    let w: usize = it.next().unwrap().parse().unwrap();
    let mut a: Vec<usize> = it.map(|s| s.parse().unwrap()).collect();
    a.sort();
    let mut dp = vec![vec![ModInt::one(); 1]; 1];
    for (i, a) in a.into_iter().rev().enumerate() {
        let mut next = vec![vec![ModInt::zero(); w + 1]; min(i + 2, n + 1 - i)];
        for (x, dp) in dp.into_iter().enumerate() {
            if a * x > w {
                break;
            }
            for (y, v) in dp.into_iter().enumerate() {
                if v.0 == 0 {
                    continue;
                }
                if let Some(p) = get_mut!(next; x + 1, y + a) {
                    *p += v;
                }
                if let Some(p) = get_mut!(next; x, y) {
                    *p += v * ModInt((y - a * x + x) as u32);
                }
                let u = v * ModInt((2 * x) as u32);
                for d in 1..(a + 1) {
                    if let Some(p) = get_mut!(next; x, y + d) {
                        *p += u;
                    }
                }
                if x > 1 {
                    let u = v * ModInt((x * (x - 1)) as u32);
                    let mut add = ModInt::zero();
                    for d in (1..(a + 1)).rev() {
                        add += u;
                        if let Some(p) = get_mut!(next; x - 1, y + d) {
                            *p += add;
                        }
                    }
                }
            }
        }
        dp = next;
    }
    let ans = dp[1][w];
    println!("{}", ans);
}

fn main() {
    run();
}