Ascending and Descending

There are cards numbered 1 to 9 lying on a table. It is known that they can be arranged in \(9!\) ways in a line. In how many ways can these cards be arranged (in a line) so that there are no 4 cards that are in ascending or descending order?

Example: In the sequence of numbers 3-2-1-6-5-4-9-8-7, there are no 4 cards in either ascending or descending order (from left to right).

Details and assumptions:

  1. The 4 cards need not be adjacent to each other.
  2. The numbers are read from left to right in the line.
  3. Source: numberphile

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