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How many unit paths are there in $\Bbb R^3$ from $(0,0,0)$ to $(n,n,n)$ that never pass below the plane $y=x$? This means that for every point $(a,b,c)$ on the path we have $a\le b$.

Brian M. Scott
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KGT
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1 Answers1

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Hint: Do you know how to do it in $\Bbb R^2$? See the Catalan numbers. Then pick $n$ places of the path of $3n$ to put steps in the third axis. Why does this work?

Ross Millikan
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  • I think that for R2 you would do 2n choose n to get the total number off paths and then do 2n choose n-1 to get the number of paths that go below and subtract them. – KGT Oct 27 '16 at 21:06
  • I was asking why my approach works to move from 2D to 3D – Ross Millikan Oct 27 '16 at 21:15