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domotorp
domotorp
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The number of Dyck paths in a square is well-known to equal the catalan numbers: http://mathworld.wolfram.com/DyckPath.html But what if, instead of a square, we ask the same question with a rectangle? If one of its sides is a multiple of the other, then again there is a nice formula for the number of paths below the diagonal, but is there a nice formula in general? What is the number of paths from the lower-left corner of a rectangle with side lengths a and b to its upper-right corner staying below the diagonal (except for its endpoint)? I am also interested in asymptotics.

The number of Dyck paths in a square is well-known to equal the catalan numbers: http://mathworld.wolfram.com/DyckPath.html But what if, instead of a square, we ask the same question with a rectangle? If one of its sides is a multiple of the other, then again there is a nice formula for the number of paths below the diagonal, but is there a nice formula in general? I am also interested in asymptotics.

The number of Dyck paths in a square is well-known to equal the catalan numbers: http://mathworld.wolfram.com/DyckPath.html But what if, instead of a square, we ask the same question with a rectangle? If one of its sides is a multiple of the other, then again there is a nice formula for the number of paths below the diagonal, but is there a nice formula in general? What is the number of paths from the lower-left corner of a rectangle with side lengths a and b to its upper-right corner staying below the diagonal (except for its endpoint)? I am also interested in asymptotics.

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domotorp
domotorp
  • 19.1k
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  • 59
  • 128

Dyck paths on rectangles

The number of Dyck paths in a square is well-known to equal the catalan numbers: http://mathworld.wolfram.com/DyckPath.html But what if, instead of a square, we ask the same question with a rectangle? If one of its sides is a multiple of the other, then again there is a nice formula for the number of paths below the diagonal, but is there a nice formula in general? I am also interested in asymptotics.