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.

Key Information

Named after	Theodore Motzkin
Publication year	1948
Author of publication	Theodore Motzkin
No. of known terms	infinity
Formula	see Properties
First terms	1, 1, 2, 4, 9, 21, 51
OEIS index	A001006 Motzkin
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The Motzkin numbers ${\displaystyle M_{n}}$ for ${\displaystyle n=0,1,\dots }$ form the sequence:

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ... (sequence A001006 in the OEIS)

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):

The Motzkin numbers satisfy the recurrence relations

${\displaystyle M_{n}=M_{n-1}+\sum _{i=0}^{n-2}M_{i}M_{n-2-i}={\frac {2n+1}{n+2}}M_{n-1}+{\frac {3n-3}{n+2}}M_{n-2}.}$

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

${\displaystyle M_{n}=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}C_{k},}$

and inversely,^[1]

${\displaystyle C_{n+1}=\sum _{k=0}^{n}{\binom {n}{k}}M_{k}}$

This gives

${\displaystyle \sum _{k=0}^{n}C_{k}=1+\sum _{k=1}^{n}{\binom {n}{k}}M_{k-1}.}$

The generating function ${\displaystyle m(x)=\sum _{n=0}^{\infty }M_{n}x^{n}}$ of the Motzkin numbers satisfies

${\displaystyle x^{2}m(x)^{2}+(x-1)m(x)+1=0}$

and is explicitly expressed as

${\displaystyle m(x)={\frac {1-x-{\sqrt {1-2x-3x^{2}}}}{2x^{2}}}.}$

An integral representation of Motzkin numbers is given by

${\displaystyle M_{n}={\frac {2}{\pi }}\int _{0}^{\pi }\sin(x)^{2}(2\cos(x)+1)^{n}dx}$.

They have the asymptotic behaviour

${\displaystyle M_{n}\sim {\frac {1}{2{\sqrt {\pi }}}}\left({\frac {3}{n}}\right)^{3/2}3^{n},~n\to \infty }$.

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

2, 127, 15511, 953467954114363 (sequence A092832 in the OEIS)

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.

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

• 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, Series A, 23 (3): 291–301, 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

• Weisstein, Eric W. "Motzkin Number". MathWorld.