Motzkin number


In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle. The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory.
The Motzkin numbers for form the sequence:

Examples

The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle :
The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle :

Properties

The Motzkin numbers satisfy the recurrence relations
The Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers:
The generating series of the Motzkin numbers satisfies
A Motzkin prime is a Motzkin number that is prime., four such primes are known:

Combinatorial interpretations

The Motzkin number for is also the number of positive integer sequences of length in which the opening and ending elements are either 1 or 2, and the difference between any two consecutive elements is −1, 0 or 1. Equivalently, the Motzkin number for is the number of positive integer sequences of length in which the opening and ending elements are 1, and the difference between any two consecutive elements is −1, 0 or 1.
Also, the Motzkin number for gives the number of routes on the upper right quadrant of a grid from coordinate to coordinate in steps if one is allowed to move only to the right at each step but forbidden from dipping below the = 0 axis.
For example, the following figure shows the 9 valid Motzkin paths from to :
There are at least fourteen different manifestations of Motzkin numbers in different branches of mathematics, as enumerated by in their survey of Motzkin numbers.
showed that vexillary involutions are enumerated by Motzkin numbers.