User:Richard Pinch/sandbox-12

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 $$C_n = \frac{1}{n+1}\binom{2n}{n} \ .$$

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:

• Bernoulli excursion
• Dyck paths
• Parenthesised sequences; words of the Dyck language
• Complete binary rooted plane trees

Poisson ratio

The ratio of longitudinal extension to lateral compression when an elastic substance is put under tension.

References

• Horace Lamb, "Statics", Cambridge University Press (1960)