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This proof uses the triangulation definition of Catalan numbers to establish a relation between C n and C n+1. Given a polygon P with n + 2 sides and a triangulation, mark one of its sides as the base, and also orient one of its 2n + 1 total edges. There are (4n + 2)C n such marked triangulations for a given base.
This number is given by the 5th Catalan number. It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex n-gon by non-intersecting diagonals is the (n−2)nd Catalan number, which equals
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves.
The number of vertices in K n+1 is the n-th Catalan number (right diagonal in the triangle). The number of facets in K n +1 (for n ≥2) is the n -th triangular number minus one (second column in the triangle), because each facet corresponds to a 2- subset of the n objects whose groupings form the Tamari lattice T n , except the 2-subset that ...
The counting sequence of a combinatorial class is the sequence of the numbers of ... are both counted by the Catalan numbers, ... a triangulation can be ...
The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the n th Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves.
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is