In how many ways can a convex polygon with \((n+2)\) sides be divided into \(n\) non-overlapping triangles (or perhaps the number of ways that \(2\cdot n\) people at a table can introduce themselves by shaking hands without reaching across in front of someone else!) ⇓

Leonhard Euler asked the first question in 1751, leading to the Catalan and Hyper-Catalan numbers! More?

Tap (+) or (-) for n to select your polygon, and use the variants buttons to see some of the different possible arrangements, then try entering some yourself! Use the Variant Type button to switch between Catalan and Hyper-Catalan numbers. ⇓

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\[n\]							▶
		0		0	12
\[Variants\]						▶
			-80			0	42
r				▶
		0		0	50

( - ) \(n\) ( + ) ( - ) \(Variants\) ( + ) ( - ) \( Variant \ Type \) ( + )

Variant Type: Catalan

Plot my variant ⇑ RESET

App generated by Geometry Expressions

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