Labeled Fuss-Catalan Polygon Divisions
Labeled Fuss-Catalan Polygon Divisions
The Catalan numbers have many interpretations in combinatorics, two of them being the number of ways to insert parentheses into a sequence of numbers so that they are grouped into pairs—for example, ((12)3) and (1(23))—and the number of ways to divide a polygon into triangles.
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The Fuss–Catalan numbers are a generalization of the Catalan numbers and have many analogous interpretations. For example, two interpretations of the Fuss–Catalan numbers are the number of ways to insert parentheses in a sequence of numbers so that they are grouped into triples—for example, ((123)45), (1(234)5) and (12(345))—and the number of ways to divide a polygon with an even number of sides into quadrilaterals. This Demonstration shows the equivalence of these two Fuss–Catalan interpretations by labeling each polygon division with the corresponding parenthesized expression.
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