Motzkin number

In mathematics, the $n$th Motzkin number is the number of different ways of drawing non-intersecting chords between $n$ points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory.

The Motzkin numbers $$M_n$$ for $$n = 0, 1, \dots$$ form the sequence:


 * 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...

Examples
The following figure shows the 9 ways to draw non-intersecting chords between 4 points on a circle ($M_{4} = 9$):


 * [[Image:MotzkinChords4.svg]]

The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle ($M_{5} = 21$):


 * [[Image:MotzkinChords5.svg]]

Properties
The Motzkin numbers satisfy the recurrence relations


 * $$M_{n}=M_{n-1}+\sum_{i=0}^{n-2}M_iM_{n-2-i}=\frac{2n+1}{n+2}M_{n-1}+\frac{3n-3}{n+2}M_{n-2}.$$

The Motzkin numbers can be expressed in terms of binomial coefficients and Catalan numbers:


 * $$M_n=\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{2k} C_k,$$

and inversely,


 * $$C_{n+1}=\sum_{k=0}^{n} \binom{n}{k} M_k$$

This gives


 * $$\sum_{k=0}^{n}C_{k} = 1 + \sum_{k=1}^{n} \binom{n}{k} M_{k-1}.$$

The generating function $$m(x) = \sum_{n=0}^\infty M_n x^n$$ of the Motzkin numbers satisfies
 * $$x^2 m(x)^2 + (x - 1) m(x) + 1 = 0$$

and is explicitly expressed as
 * $$m(x) = \frac{1-x-\sqrt{1-2x-3x^2}}{2x^2}.$$

An integral representation of Motzkin numbers is given by
 * $$M_{n}=\frac{2}{\pi}\int_0^\pi \sin(x)^2(2\cos(x)+1)^n dx$$.

They have the asymptotic behaviour
 * $$M_{n}\sim \frac{1}{2 \sqrt{\pi}}\left(\frac{3}{n}\right)^{3/2} 3^n,~ n \to \infty$$.

A Motzkin prime is a Motzkin number that is prime. Four such primes are known:


 * 2, 127, 15511, 953467954114363

Combinatorial interpretations
The Motzkin number for $n$ is also the number of positive integer sequences of length $n &minus; 1$ in which the opening and ending elements are either 1 or 2, and the difference between any two consecutive elements is &minus;1, 0 or 1. Equivalently, the Motzkin number for $n$ is the number of positive integer sequences of length $n + 1$ in which the opening and ending elements are 1, and the difference between any two consecutive elements is &minus;1, 0 or 1.

Also, the Motzkin number for $n$ gives the number of routes on the upper right quadrant of a grid from coordinate (0, 0) to coordinate ($n$, 0) in $n$ steps if one is allowed to move only to the right (up, down or straight) at each step but forbidden from dipping below the $y$ = 0 axis.

For example, the following figure shows the 9 valid Motzkin paths from (0, 0) to (4, 0):


 * [[Image:Motzkin4.svg]]

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.