Mixed Up Cows

Each of Farmer John's $N$ cows ($4 \le N \le 16$) has a unique serial number $S_i$ ($1 \le S_i \le 25000$).

The cows line up to be milked in an order called 'Mixed Up'. A lineup is 'Mixed Up' if, in the sequence of serial numbers formed by the line, every pair of consecutive cows has serial numbers that differ by more than $K$ ($1 \le K \le 3400$). For example, if $N = 6$ and $K = 1$, then the order $1, 3, 5, 2, 6, 4$ is 'Mixed Up', but $1, 3, 6, 5, 2, 4$ is not, because the consecutive numbers $5$ and $6$ differ by only $1$.

How many different ways can the $N$ cows be lined up 'Mixed Up'?

• Line 1: Two space-separated integers $N$ and $K$.
• Lines 2 to $N+1$: Line $i+1$ contains a single integer $S_i$, the serial number of cow $i$.

• Line 1: A single integer, the number of ways the $N$ cows can be lined up 'Mixed Up'. The answer is guaranteed to fit in a 64-bit integer.

For the sample, the $2$ possible 'Mixed Up' lineups are shown below, where each line lists the cows' serial numbers in order.

3 1 4 2
2 4 1 3