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 very diverse applications in geometry, combinatorics and number theory.

Motzkin numbers ${\displaystyle M_{n}}$ for ${\displaystyle n=0,1,\dots }$ form the sequence:

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.

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