Question

A Motzkin path of length n is a sequence of points (0, 0) = P0, P1, ..., Pn = (n, 0) where Pi - Pi-1 ∈ {(1, 1), (1, -1), (1, 0)} and the y-coordinate of Pi is nonnegative for all i. Below are all the Motzkin paths from (0, 0) to (4, 0). (a) Find a recursive construction for the set of all Motzkin paths. (b) Use the construction you found in the previous part to find the generating function for Motzkin paths of any length.

Name: a motzkin path of length n is a sequence of points 0 0 p0 p1 pn n 0 where pi pi 1 1 1 1 1 1 0 and the y coordinate of pi is nonnegative for all i below are all the motzkin paths from 0 0 to 86317
Uploaded: 2022-12-13T12:05:45-08:00
Duration: 4 min 10 s
Channel: Sri K
Description: a motzkin path of length n is a sequence of points 0 0 p0 p1 pn n 0 where pi pi 1 1 1 1 1 1 0 and the y coordinate of pi is nonnegative for all i below are all the motzkin paths from 0 0 to 86317

          A Motzkin path of length n is a sequence of points (0, 0) = P0, P1, ..., Pn = (n, 0) where Pi - Pi-1 ∈ {(1, 1), (1, -1), (1, 0)} and the y-coordinate of Pi is nonnegative for all i. Below are all the Motzkin paths from (0, 0) to (4, 0).

(a) Find a recursive construction for the set of all Motzkin paths.
(b) Use the construction you found in the previous part to find the generating function for Motzkin paths of any length.

a motzkin path of length n is a sequence of points 0 0 p0 p1 pn n 0 where pi pi 1 1 1 1 1 1 0 and the y coordinate of pi is nonnegative for all i below are all the motzkin paths from 0 0 to 86317

Added by Catherine B.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

12/13/2022

Step 1

The last step can be one of three possibilities: (1, 1), (1, -1), or (1, 0). If the last step is (1, 1), then the previous n-1 steps must form a Motzkin path of length n-1 that ends at a point with nonnegative y-coordinate. This gives us a recursive formula for Show more…

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A Motzkin path of length n is a sequence of points (0, 0) = P0, P1, ..., Pn = (n, 0) where Pi - Pi-1 ∈ {(1, 1), (1, -1), (1, 0)} and the y-coordinate of Pi is nonnegative for all i. Below are all the Motzkin paths from (0, 0) to (4, 0). (a) Find a recursive construction for the set of all Motzkin paths. (b) Use the construction you found in the previous part to find the generating function for Motzkin paths of any length.