Dissection of Polygon into Triangles with Chords counting Isometries

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

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Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $42$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $42$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Catalan numbers