Motzkin number

Summary

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.

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

The Motzkin numbers for 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)

ExamplesEdit

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):

 

PropertiesEdit

The Motzkin numbers satisfy the recurrence relations

 

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

 

and inversely,[1]

 

The generating function   of the Motzkin numbers satisfies

 

and is explicitly expressed as

 

An integral representation of Motzkin numbers is given by

 .

They have the asymptotic behaviour

 .

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 interpretationsEdit

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 alsoEdit

ReferencesEdit

  1. ^ 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
  • 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 linksEdit