Motzkin number

☆ Save On Wikipedia ↗
Motzkin number
Named afterTheodore Motzkin
Publication year1948
Author of publicationTheodore Motzkin
No. of known termsinfinity
Formulasee Properties
First terms1, 1, 2, 4, 9, 21, 51
OEIS index

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 {\displaystyle M_{n}} {\displaystyle M_{n}} for n = 0 , 1 , … {\displaystyle n=0,1,\dots } {\displaystyle 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 (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 + ∑ i = 0 n − 2 M i M n − 2 − i = 2 n + 1 n + 2 M n − 1 + 3 n − 3 n + 2 M n − 2 . {\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}.} {\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:

M n = ∑ k = 0 ⌊ n / 2 ⌋ ( n 2 k ) C k , {\displaystyle M_{n}=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}C_{k},} {\displaystyle M_{n}=\sum _{k=0}^{\lfloor n/2\rfloor }{\binom {n}{2k}}C_{k},}

and inversely,[1]

C n + 1 = ∑ k = 0 n ( n k ) M k {\displaystyle C_{n+1}=\sum _{k=0}^{n}{\binom {n}{k}}M_{k}} {\displaystyle C_{n+1}=\sum _{k=0}^{n}{\binom {n}{k}}M_{k}}

This gives

∑ k = 0 n C k = 1 + ∑ k = 1 n ( n k ) M k − 1 . {\displaystyle \sum _{k=0}^{n}C_{k}=1+\sum _{k=1}^{n}{\binom {n}{k}}M_{k-1}.} {\displaystyle \sum _{k=0}^{n}C_{k}=1+\sum _{k=1}^{n}{\binom {n}{k}}M_{k-1}.}

The generating function m ( x ) = ∑ n = 0 ∞ M n x n {\displaystyle m(x)=\sum _{n=0}^{\infty }M_{n}x^{n}} {\displaystyle 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 {\displaystyle x^{2}m(x)^{2}+(x-1)m(x)+1=0} {\displaystyle x^{2}m(x)^{2}+(x-1)m(x)+1=0}

and is explicitly expressed as

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

An integral representation of Motzkin numbers is given by

M n = 2 π ∫ 0 π sin ⁡ ( x ) 2 ( 2 cos ⁡ ( x ) + 1 ) n d x {\displaystyle M_{n}={\frac {2}{\pi }}\int _{0}^{\pi }\sin(x)^{2}(2\cos(x)+1)^{n}dx} {\displaystyle M_{n}={\frac {2}{\pi }}\int _{0}^{\pi }\sin(x)^{2}(2\cos(x)+1)^{n}dx}.

They have the asymptotic behaviour

M n ∼ 1 2 π ( 3 n ) 3 / 2 3 n ,   n → ∞ {\displaystyle M_{n}\sim {\frac {1}{2{\sqrt {\pi }}}}\left({\frac {3}{n}}\right)^{3/2}3^{n},~n\to \infty } {\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)

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

References

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