Motzkin number

In mathematics, the nth 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 (M₄ = 9):

The following figure shows the 21 ways to draw non-intersecting chords between 5 points on a circle (M₅ = 21):

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,^[1]

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 − 1 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 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 −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):

There are at least fourteen different manifestations of Motzkin numbers in different branches of mathematics, as enumerated by Donaghey & Shapiro (1977) in their survey of Motzkin numbers. Guibert, Pergola & Pinzani (2001) showed that vexillary involutions are enumerated by Motzkin numbers.

See also

• Telephone number which represent the number of ways of drawing chords if intersections are allowed
• Delannoy number
• Narayana number
• Schröder number

References

1. ^ Yi Wang and Zhi-Hai Zhang (2015). "Combinatorics of Generalized Motzkin Numbers". Journal of Integer Sequences. ' (18)

• Bernhart, Frank R. (1999), "Catalan, Motzkin, and Riordan numbers", Discrete Mathematics. 204 (1–3): 73–112, doi:10.1016/S0012-365X(99)00054-0
• Donaghey, R. & Shapiro, L. W. (1977), "Motzkin numbers", Journal of Combinatorial Theory. 23 (3): 291–301, Series A, doi:10.1016/0097-3165(77)90020-6. MR 0505544
• Guibert, O.; Pergola, E.; Pinzani, R. (2001), "Vexillary involutions are enumerated by Motzkin numbers", Annals of Combinatorics. 5 (2): 153–174, doi:10.1007/PL00001297. ISSN 0218-0006. MR 1904383. S2CID 123053532
• Motzkin, T. S. (1948), "Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products", Bulletin of the American Mathematical Society. 54 (4): 352–360, doi:10.1090/S0002-9904-1948-09002-4