Motzkin number

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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.

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 OEIS)

Examples[edit]

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

MotzkinChords4.svg

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

MotzkinChords5.svg

Properties[edit]

Motzkin numbers satisfy the recurrence relation:

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}.

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 OEIS)

Combinatorial interpretations[edit]

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):

Motzkin4.svg

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.

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