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Difference between revisions of "User:Richard Pinch/sandbox-12"

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(Start article: Dyck path)
 
m (→‎Dyck path: typo)
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=Dyck path=
 
=Dyck path=
A [[lattice path]] on the square lattice from the origin $(0m0)$ to some point $(n,n)$ consisting of $2n$ stepsn of the form $N : (x,y) \rightarrow (x,y+1)$ and $E : (x,y) \rightarrow (x+1,y)$ with the property that the path
+
A [[lattice path]] on the square lattice from the origin $(0,0)$ to some point $(n,n)$ consisting of $2n$ steps of the form $N : (x,y) \rightarrow (x,y+1)$ and $E : (x,y) \rightarrow (x+1,y)$ with the property that the path
 
never passes below the line $y=x$.
 
never passes below the line $y=x$.
  

Revision as of 14:58, 26 December 2017

Dyck path

A lattice path on the square lattice from the origin $(0,0)$ to some point $(n,n)$ consisting of $2n$ steps of the form $N : (x,y) \rightarrow (x,y+1)$ and $E : (x,y) \rightarrow (x+1,y)$ with the property that the path never passes below the line $y=x$.

The number of Dyck paths of length $2n$ is given by the $n$-th Catalan number $$ C_n = \frac{1}{n+1}\binom{2n}{n} \ . $$

References

How to Cite This Entry:
Richard Pinch/sandbox-12. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-12&oldid=42596