Problem Description

A sequence of non-negative integers a1, a2, ..., an of length n is called a wool sequence if and only if there exists two integers l and r (1 ≤ l ≤ r ≤ n) such that . In other words each wool sequence contains a subsequence of consecutive elements with xor equal to 0.

The expression  means applying the operation of a bitwise xor to numbers x and y. The given operation exists in all modern programming languages, for example, in languages C++ and Java it is marked as "^", in Pascal — as "xor".

In this problem you are asked to compute the number of sequences made of n integers from 0 to 2m - 1 that are not a wool sequence. You should print this number modulo 1000000009 (109 + 9).

Input

The only line of input contains two space-separated integers n and m (1 ≤ n, m ≤ 105).

Output

Print the required number of sequences modulo 1000000009 (109 + 9) on the only line of output.

Examples

Input

3 2

Output

6

Note

Sequences of length 3 made of integers 0, 1, 2 and 3 that are not a wool sequence are (1, 3, 1), (1, 2, 1), (2, 1, 2), (2, 3, 2), (3, 1, 3) and (3, 2, 3).

题意：给出 n、m 两个数，现在要在 0~2^m-1 个数中取可重复的 n 个数，使得组成的序列异或和为 0，问这 2^m-1 个数中组成的序列有多少个满足要求

思路：

设一个前缀数组，使得 sum[i]=sum[i-1]^a[i]，那么 a[i]=sum[i-1]^sum[i]

假设 0~2^m-1 不是一个满足要求的序列，那么 a[l]^a[l+1]^...^a[r]!=0，即有：sum[l-1]^sum[r]!=0

由于 l<=r，因此 sum 数组中的所有数需要两两不同

又因为 a[i]=sum[i-1]^sum[i]，因此 sum[i] 的取值在 [1,2^m]

因此，问题就变成求有多少个长度为 n 的数组，且数组中的数在 [1,2^m] 之间且不相同

Source Program

#include<iostream> #include<cstdio> #include<cstdlib> #include<string> #include<cstring> #include<cmath> #include<ctime> #include<algorithm> #include<utility> #include<stack> #include<queue> #include<vector> #include<set> #include<map> #include<bitset> #define EPS 1e-9 #define PI acos(-1.0) #define INF 0x3f3f3f3f #define LL long long const int MOD = 1E9+9; const int N = 1000000+5; const int dx[] = {-1,1,0,0,-1,-1,1,1}; const int dy[] = {0,0,-1,1,-1,1,-1,1}; using namespace std;int main(){LL n,m;scanf("%lld%lld",&n,&m);LL mul=1,res=1;for(int i=1;i<=m;i++)mul=(mul*2)%MOD;for(int i=1;i<=n;i++)res=(res*(mul-i))%MOD;printf("%lld\n",res);return 0; }

 

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