Partitioning

Time limit0.5sMemory limit512 MB

Summary

Decide whether the set {0,...,2^N-1} splits into two subsets of sizes K and 2^N-K, each closed under bitwise OR, and print one split if it exists.

Level: Medium7 of 10
Topics: Bit manipulation, Combinatorics, Math, Brute force
Solved: No attempts yet

Problem

For a set $S$ , call $S$ a good set when $a,b \in S \Longrightarrow (a \mid b) \in S$ holds, and only then. Here $\mid$ denotes bitwise-or. For example, $\{1, 3\}$ is a good set, because $(1 \mid 1) = 1$ , $(1 \mid 3) = 3$ , and $(3 \mid 3) = 3$ are all elements of $\{1, 3\}$ . However, $(1 \mid 2) = 3$ , so $\{1, 2\}$ is not a good set.

For a subset $A$ of the universe $U = \{0, 1, \cdots, 2^N - 1\}$ , call $A$ an $N, K$ -partition when $A$ has $K$ elements and both $A$ and $U \setminus A$ are good sets, and only then.

Given $N$ and $K$ , write a program that determines whether an $N, K$ -partition exists and, if it does, prints an example of one.

Input

The first line gives the natural numbers $N$ and $K$ separated by a space.

Output

If no $N, K$ -partition exists, print NO on the first line.

If an $N, K$ -partition exists, print YES on the first line. On the second line, print the elements of an $N, K$ -partition in increasing order.

If multiple $N, K$ -partitions exist, printing any one of them is accepted.

Constraints

Every input satisfies the following conditions.

$1 \le N \le 12$
$1 \le K \le 2^N$

Examples4

Example 1
Input
```
1 1
```
Expected output
```
YES
0
```
Example 2
Input
```
2 4
```
Expected output
```
YES
0 1 2 3
```
Example 3
Input
```
3 3
```
Expected output
```
YES
1 4 5
```
Example 4
Input
```
3 5
```
Expected output
```
YES
0 2 3 6 7
```