Given that a mountain path is a sequence steps that never go under the x- axis (no negative points) and are sequences of NE(going up one unit and right one unit) and SE(going down one unit and right one unit). Prove that the mountain paths from point (0, 0) -> (2x, 0) are Catalan object. Note: A Catalan object is proven by one of the following (i) a bijection of a known Catalan object (ii) show it satifies a Catalan recursion (iii) directly count it and find there are (1/(x+1))*(2x choose x) of them
Given that a mountain path is a sequence steps that never go under the x- axis (no negative points) and are sequences of NE(going up one unit and right one unit) and SE(going down one unit and right one unit). Prove that the mountain paths from point (0, 0) -> (2x, 0) are Catalan object. Note: A Catalan object is proven by one of the following (i) a bijection of a known Catalan object (ii) show it satifies a Catalan recursion (iii) directly count it and find there are (1/(x+1))*(2x choose x) of them
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given that a mountain path is a sequence steps that never go under the x- axis (no negative points) and are sequences of NE(going up one unit and right one unit) and SE(going down one unit and right one unit). Prove that the mountain paths from point (0, 0) -> (2x, 0) are Catalan object.
Note: A Catalan object is proven by one of the following
(i) a bijection of a known Catalan object
(ii) show it satifies a Catalan recursion
(iii) directly count it and find there are (1/(x+1))*(2x choose x) of them
Each mountain path for x = 3 are shown in attached photo:
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