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In mathematics, a Motzkin number for a given number n (named after Theodore Motzkin) is the number of different ways of drawing non-intersecting chords on a circle between n points. The Motzkin numbers have very diverse applications in geometry, combinatorics and number theory. The recurrence relation is:

$$M_{n+1}=M_n+\sum_{i=0}^{n-1}M_iM_{n-1-i}=\frac{2n+3}{n+3}M_n+\frac{3n}{n+3}M_{n-1}$$

The first few Motzkin numbers are (sequence A001006 in OEIS):

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

The following figure (*) shows the 9 ways to draw non-intersecting chords between 4 points on a circle.

The following figure (*) shows the 21 ways to draw non-intersecting chords between 5 points on a circle.

A Motzkin prime is a Motzkin number that is prime. As of October 2007, four such primes are known (sequence A092832 in OEIS):

2, 127, 15511, 953467954114363

The Motzkin number for n is also the number of positive integer sequences n−1 long 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.

Also on the upper right quadrant of a grid, the Motzkin number for n gives the number of routes from coordinate (0, 0) to coordinate (n, 0) on 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.

Delannoy number
Narayana number
Schröder number

References

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, MR0505544
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, MR1904383
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