Battle of Round Mountain

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, ... (sequence A001006 in the OEIS)

Examples

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

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

Properties

The Motzkin numbers satisfy the recurrence relations

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:

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. As of 2019, only four such primes are known:

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

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.

References

^ Yi Wang and Zhi-Hai Zhang (2015). "Combinatorics of Generalized Motzkin Numbers" (PDF). 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, 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

External links

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

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

[1]

Examples

Properties

Combinatorial interpretations

See also

References

External links