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There are $8$ pawns next to each other and by using any size of discs you need to cover all pawns. The conditions while putting discs is that:

  • Every disc and every pawn is called an object.
  • There has to be two objects in a disc.
  • The pawns in the disc is not being counted as different objects for outer disc.

So, how many different way to put these discs is there?

For example if this question was asked for $4$ pawns the answer would be $5$ as shown below:

enter image description here

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You're asking for the

eighth Catalan number, which equals 429. Many problems with this sort of "nested" structure end up being about Catalan numbers.

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