Motzkin number

In mathematics, a Motzkin number for a given number $n$ 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, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829, ... (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_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.

A Motzkin prime is a Motzkin number that is prime. As of October 2013, 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

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
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.

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