6. 2892.

From the problem, we know that $b$ and $d$ are the maximum and minimum numbers, respectively, and $2 \leqslant b-d \leqslant 9$.

Let $b-d=k$, at this point, there are $10-k$ ways to choose $(b, d)$, and $a$, $c$, and $e$ each have $k-1$ ways to be chosen, i.e., $(a, c, e)$ has $(k-1)^{3}$ combinations. Therefore, the number of numbers that conform to the sine rule is $(k-1)^{3}(10-k)$.
Thus, the number of numbers that conform to the sine rule is
$$
\sum_{k=2}^{9}(k-1)^{3}(10-k)=2892 .
$$
6. A five-digit number abcde satisfies:
$$
ac>d, dd, b>e \text {. }
$$

For example, 34201, 49412. If the digits of the number change with the position in a way similar to the monotonicity of a sine function over one period, then the five-digit number is said to conform to the "sine rule." The number of five-digit numbers that conform to the sine rule is $\qquad$.
