Problem:
Score : 1100 points

Problem Statement

There are 2^N players, numbered 1, 2, ..., 2^N.
They decided to hold a tournament.

The tournament proceeds as follows:

Choose a permutation of 1, 2, ..., 2^N: p_1, p_2, ..., p_{2^N}.

The players stand in a row in the order of Player p_1, Player p_2, ..., Player p_{2^N}.

Repeat the following until there is only one player remaining in the row:

Play the following matches: the first player in the row versus the second player in the row, the third player versus the fourth player, and so on. The players who lose leave the row. The players who win stand in a row again, preserving the relative order of the players.

The last player who remains in the row is the champion.

It is known that, the result of the match between two players can be written as follows, using M integers A_1, A_2, ..., A_M given as input:

When y = A_i for some i, the winner of the match between Player 1 and Player y (2 \leq y \leq 2^N) will be Player y.

When y \neq A_i for every i, the winner of the match between Player 1 and Player y (2 \leq y \leq 2^N) will be Player 1.

When 2 \leq x < y \leq 2^N, the winner of the match between Player x and Player y will be Player x.

The champion of this tournament depends only on the permutation p_1, p_2, ..., p_{2^N} chosen at the beginning.
Find the number of permutation p_1, p_2, ..., p_{2^N} chosen at the beginning of the tournament that would result in Player 1 becoming the champion, modulo 10^9 + 7.

Constraints

1 \leq N \leq 16

0 \leq M \leq 16

2 \leq A_i \leq 2^N (1 \leq i \leq M)

A_i < A_{i + 1} (1 \leq i < M)

Input

Input is given from Standard Input in the following format:

N M
A_1 A_2 ... A_M

Output

Print the answer.

Sample Input 1

2 1
3

Sample Output 1

8

Examples of p that satisfy the condition are: [1, 4, 2, 3] and [3, 2, 1, 4]. Examples of p that do not satisfy the condition are: [1, 2, 3, 4] and [1, 3, 2, 4].

Sample Input 2

4 3
2 4 6

Sample Output 2

0

Sample Input 3

3 0

Sample Output 3

40320

Sample Input 4

3 3
3 4 7

Sample Output 4

2688

Sample Input 5

16 16
5489 5490 5491 5492 5493 5494 5495 5497 18993 18995 18997 18999 19000 19001 19002 19003

Sample Output 5

816646464

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::input;

// modint {{{
mod modint {
    pub trait ModValue:
        std::ops::Add<Output = Self>
        + std::ops::Sub<Output = Self>
        + std::ops::Mul<Output = Self>
        + std::ops::Div<Output = Self>
        + std::ops::Rem<Output = Self>
        + std::ops::AddAssign
        + std::ops::SubAssign
        + std::ops::MulAssign
        + std::ops::DivAssign
        + std::ops::RemAssign
        + std::ops::Neg
        + std::cmp::Ord
        + Clone
        + Copy
        + Sized
        + std::fmt::Debug
        + std::fmt::Display
    {
    }

    impl<
            Value: std::ops::Add<Output = Self>
                + std::ops::Sub<Output = Self>
                + std::ops::Mul<Output = Self>
                + std::ops::Div<Output = Self>
                + std::ops::Rem<Output = Self>
                + std::ops::AddAssign
                + std::ops::SubAssign
                + std::ops::MulAssign
                + std::ops::DivAssign
                + std::ops::RemAssign
                + std::ops::Neg
                + std::cmp::Ord
                + Clone
                + Copy
                + Sized
                + std::fmt::Debug
                + std::fmt::Display,
        > ModValue for Value
    {
    }

    pub trait Minfo: Clone + Copy + std::fmt::Debug {
        type Value: ModValue;

        fn modulus() -> Self::Value;

        fn zero() -> Self::Value;

        fn one() -> Self::Value;
    }

    #[derive(Clone, Copy, Debug)]
    pub struct Mint<Mod: Minfo>(Mod::Value);

    impl<Mod: Minfo> Mint<Mod> {
        #[inline]
        #[allow(dead_code)]
        fn is_within_the_range(x: Mod::Value) -> bool {
            Mod::zero() <= x && x < Mod::modulus()
        }

        #[inline]
        #[allow(dead_code)]
        fn normalize(x: Mod::Value) -> Mod::Value {
            if Self::is_within_the_range(x) {
                x
            } else {
                let x = x % Mod::modulus();
                if x < Mod::zero() {
                    x + Mod::modulus()
                } else {
                    x
                }
            }
        }

        #[allow(dead_code)]
        pub fn zero() -> Self {
            Self(Mod::zero())
        }

        #[allow(dead_code)]
        pub fn one() -> Self {
            Self(Mod::one())
        }

        #[allow(dead_code)]
        pub fn modulus() -> Mod::Value {
            Mod::modulus()
        }

        #[allow(dead_code)]
        pub fn is_zero(&self) -> bool {
            self.value() == Mod::zero()
        }

        #[allow(dead_code)]
        pub fn from_value(x: Mod::Value) -> Self {
            Self(Self::normalize(x))
        }

        #[allow(dead_code)]
        pub fn value(&self) -> Mod::Value {
            self.0
        }

        #[allow(dead_code)]
        pub fn inv(&self) -> Self {
            if self.is_zero() {
                panic!("inv(0)");
            }
            Self(Self::raw_inv(self.value()))
        }

        #[allow(dead_code)]
        pub fn pow(&self, b: u64) -> Self {
            Self(Self::raw_pow(self.value(), b))
        }

        fn raw_add(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            let z = x + y;
            if Mod::modulus() <= z {
                z - Mod::modulus()
            } else {
                z
            }
        }

        fn raw_sub(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            let z = x - y;
            if z < Mod::zero() {
                z + Mod::modulus()
            } else {
                z
            }
        }

        fn raw_neg(x: Mod::Value) -> Mod::Value {
            if x == Mod::zero() {
                Mod::zero()
            } else {
                Mod::modulus() - x
            }
        }

        fn raw_mul(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            x * y % Mod::modulus()
        }

        fn raw_div(x: Mod::Value, y: Mod::Value) -> Mod::Value {
            Self::raw_mul(x, Self::raw_inv(y))
        }

        #[allow(clippy::many_single_char_names)]
        fn raw_inv(mut x: Mod::Value) -> Mod::Value {
            let mut y = Mod::modulus();
            let mut u = Mod::one();
            let mut v = Mod::zero();
            while x != Mod::zero() {
                let q = y / x;
                y -= x * q;
                v -= u * q;
                std::mem::swap(&mut x, &mut y);
                std::mem::swap(&mut u, &mut v);
            }
            Self::normalize(v)
        }

        fn raw_pow(mut a: Mod::Value, mut b: u64) -> Mod::Value {
            let mut x = Mod::one();
            while 0 < b {
                if b % 2 == 1 {
                    x = Self::raw_mul(x, a);
                }
                a = Self::raw_mul(a, a);
                b /= 2
            }
            x
        }
    }

    macro_rules! impl_biop {
        (
            $(
                $biop_trait: ident::$biop_fn: ident,
                $biop_assign_trait: ident::$biop_assign_fn: ident,
                $biop_impl: ident;
            )*
        ) => {
            $(
                impl<Mod: Minfo> std::ops::$biop_trait for Mint<Mod> {
                    type Output = Self;
                    fn $biop_fn(self, rhs: Self) -> Self::Output {
                        Mint(Self::$biop_impl(self.value(), rhs.value()))
                    }
                }
                impl<Mod: Minfo> std::ops::$biop_assign_trait for Mint<Mod>
                where
                    Self: Copy
                {
                    fn $biop_assign_fn(&mut self, rhs: Self) {
                        use std::ops::$biop_trait;
                        *self = Mint::$biop_fn(*self, rhs)
                    }
                }
            )*
        };
    }

    impl_biop! {
        Add::add, AddAssign::add_assign, raw_add;
        Sub::sub, SubAssign::sub_assign, raw_sub;
        Mul::mul, MulAssign::mul_assign, raw_mul;
        Div::div, DivAssign::div_assign, raw_div;
    }

    impl<Mod: Minfo> std::ops::Neg for Mint<Mod> {
        type Output = Self;
        fn neg(self) -> Self::Output {
            Self(Self::raw_neg(self.value()))
        }
    }

    impl<Mod: Minfo> std::cmp::PartialEq for Mint<Mod> {
        fn eq(&self, other: &Self) -> bool {
            self.value().eq(&other.value())
        }
    }
    impl<Mod: Minfo> std::cmp::Eq for Mint<Mod> {}

    impl<Mod: Minfo> std::fmt::Display for Mint<Mod> {
        fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
            write!(f, "{}", self.value())
        }
    }

    impl<Mod: Minfo> std::iter::Sum<Self> for Mint<Mod> {
        fn sum<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Mint::zero(), |acc, x| acc + x)
        }
    }

    impl<Mod: Minfo> std::iter::Product<Self> for Mint<Mod> {
        fn product<I: Iterator<Item = Self>>(iter: I) -> Self {
            iter.fold(Mint::one(), |acc, x| acc * x)
        }
    }

    #[allow(dead_code)]
    pub struct Factorial<Mod: Minfo> {
        normal: Vec<Mint<Mod>>,
        inverted: Vec<Mint<Mod>>,
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> Factorial<Mod>
    where
        Mod::Value: std::convert::TryFrom<usize, Error = std::num::TryFromIntError>,
    {
        pub fn new(n: usize) -> Self {
            use std::convert::TryFrom;
            let mut normal = vec![Mint::one(); n];
            for i in 1..n {
                normal[i] = normal[i - 1] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            let mut inverted = vec![Mint::one(); n];
            inverted[n - 1] = normal[n - 1].inv();
            for i in (1..n).rev() {
                inverted[i - 1] = inverted[i] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            Self { normal, inverted }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> Factorial<Mod> {
        fn is_empty(&self) -> bool {
            self.normal.is_empty()
        }
        fn len(&self) -> usize {
            self.normal.len()
        }
        pub fn inverted(&self, i: usize) -> Mint<Mod> {
            self.inverted[i]
        }
        pub fn falling(&self, x: usize, len: usize) -> Mint<Mod> {
            if x < len {
                Mint::zero()
            } else {
                assert!(
                    x < self.len(),
                    "The base of falling factorial should be less than the length."
                );
                self[x] * self.inverted(x - len)
            }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo, I: std::slice::SliceIndex<[Mint<Mod>]>> std::ops::Index<I> for Factorial<Mod> {
        type Output = I::Output;

        #[inline]
        fn index(&self, index: I) -> &Self::Output {
            &self.normal[index]
        }
    }

    #[allow(dead_code)]
    pub struct DoubleFactorial<Mod: Minfo> {
        normal: Vec<Mint<Mod>>,
        inverted: Vec<Mint<Mod>>,
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> DoubleFactorial<Mod>
    where
        Mod::Value: std::convert::TryFrom<usize, Error = std::num::TryFromIntError>,
    {
        pub fn new(n: usize) -> Self {
            use std::convert::TryFrom;
            let mut normal = vec![Mint::one(); n];
            for i in 2..n {
                normal[i] = normal[i - 2] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            let mut inverted = vec![Mint::one(); n];
            inverted[n - 1] = normal[n - 1].inv();
            inverted[n - 2] = normal[n - 2].inv();
            for i in (2..n).rev() {
                inverted[i - 2] = inverted[i] * Mint::from_value(Mod::Value::try_from(i).unwrap());
            }
            Self { normal, inverted }
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo> DoubleFactorial<Mod> {
        pub fn inverted(&self, i: usize) -> Mint<Mod> {
            self.inverted[i]
        }
    }

    #[allow(dead_code)]
    impl<Mod: Minfo, I: std::slice::SliceIndex<[Mint<Mod>]>> std::ops::Index<I>
        for DoubleFactorial<Mod>
    {
        type Output = I::Output;

        #[inline]
        fn index(&self, index: I) -> &Self::Output {
            &self.normal[index]
        }
    }

    #[macro_export]
    macro_rules! define_mint {
        (
            struct $modint: ident ($modinfo: ident ($modvalue: expr; $modty: ty));
        ) => {
            #[allow(dead_code)]
            #[derive(Clone, Copy, Debug)]
            pub struct $modinfo {}
            impl Minfo for $modinfo {
                type Value = $modty;
                fn modulus() -> Self::Value {
                    $modvalue
                }
                fn zero() -> Self::Value {
                    0
                }
                fn one() -> Self::Value {
                    1
                }
            }
            type $modint = Mint<$modinfo>;
        };
    }

    #[allow(dead_code)]
    #[derive(Clone, Copy, Debug)]
    pub struct Minfo1000000007 {}

    impl Minfo for Minfo1000000007 {
        type Value = i64;
        fn modulus() -> Self::Value {
            1_000_000_007
        }
        fn zero() -> Self::Value {
            0
        }
        fn one() -> Self::Value {
            1
        }
    }

    #[allow(dead_code)]
    pub type Mint1000000007 = Mint<Minfo1000000007>;

    #[allow(dead_code)]
    #[derive(Clone, Copy, Debug)]
    pub struct Minfo998244353 {}

    impl Minfo for Minfo998244353 {
        type Value = i64;
        fn modulus() -> Self::Value {
            998_244_353
        }
        fn zero() -> Self::Value {
            0
        }
        fn one() -> Self::Value {
            1
        }
    }

    #[allow(dead_code)]
    pub type Mint998244353 = Mint<Minfo998244353>;
}
// }}}

type Mint = modint::Mint1000000007;
type Minfo = modint::Minfo1000000007;

fn main() {
    input!(n: usize, m: usize,mut a: [usize; m]);
    a.iter_mut().for_each(|x| *x -= 2);
    a.push((1 << n) - 1);
    let mut dp = vec![Mint::zero(); 1 << n];
    dp[0] = Mint::one();

    let fact = modint::Factorial::<Minfo>::new((1 << n) + 1);

    let cum = |dp: &mut [Mint]| {
        for i in 0..n {
            for bs in (0..1 << n).filter(|bs| bs >> i & 1 == 0) {
                dp[bs | 1 << i] += dp[bs]
            }
        }
    };
    let adj = |dp: &mut [Mint]| {
        for i in 0..n {
            for bs in (0..1 << n).filter(|bs| bs >> i & 1 == 0) {
                dp[bs | 1 << i] -= dp[bs]
            }
        }
    };
    let translate = |dp: &mut [Mint], j: usize| {
        for (k, dp_k_mut) in dp.iter_mut().enumerate() {
            *dp_k_mut *= Mint::from_value(if k <= j { 0 } else { k - j } as i64);
        }
    };
    let translate_many = |dp: &mut [Mint], j: usize, len: usize| {
        for (k, dp_k_mut) in dp.iter_mut().enumerate() {
            *dp_k_mut *= if k <= j + len - 1 {
                Mint::zero()
            } else {
                fact.falling(k - j, len)
            };
        }
    };

    let mut j = 0;
    for x in a {
        cum(&mut dp);
        translate_many(&mut dp, j, x - j);
        j = x;
        adj(&mut dp);
        if x != (1 << n) - 1 {
            translate(&mut dp, j);
            j += 1;
        }
    }
    println!("{}", Mint::from_value(1 << n) * *dp.last().unwrap());
}