Dissection of Polygon into Triangles with Chords counting Isometries

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Theorem

The number of different ways $k$ a convex $n$-sided polygon can be divided into triangles using chords, counting reflections and rotations as different, is given for the first few $n$ as follows:

$n$ $k$
$3$ $1$
$4$ $2$
$5$ $5$
$6$ $14$
$7$ $42$
$8$ $132$
$9$ $429$
$10$ $1430$
$11$ $4862$

This sequence is A000108 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


These are the Catalan numbers.


Proof



Sources