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

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(Start article: Catalan number)
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The [[generating function]] is given by
 
The [[generating function]] is given by
 
$$
 
$$
\sum_{n=1^\infty C_n z^n = \frac{1-\sqrt{1-4z}}{2z} \ .
+
\sum_{n=1}^\infty C_n z^n = \frac{1-\sqrt{1-4z}}{2z} \ .
 
$$
 
$$
 
The Catalan numbers appear in the enumeration of a number of combinatorially defined object:
 
The Catalan numbers appear in the enumeration of a number of combinatorially defined object:

Revision as of 16:57, 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

Catalan number

The $n$-th Catalan number $$ C_n = \frac{1}{n+1}\binom{2n}{n} \ . $$ The generating function is given by $$ \sum_{n=1}^\infty C_n z^n = \frac{1-\sqrt{1-4z}}{2z} \ . $$ The Catalan numbers appear in the enumeration of a number of combinatorially defined object:

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=42598