There are n parking spots 1, 2,..., n on a one-way street. Cars 1, 2,..., n arrive in this order. Each car i has a favorite parking spot f(i). When a car arrives, it first goes to its favorite spot. If the spot is free, the car will take it, if not, it goes to the next spot. Again, if that spot is free, the car will take it, if not, the car goes to the next spot. If a car had to leave even the last spot and did not find the space, then its parking attempt has been unsuccessful. If, at the end of this procedure, all cars have a parking spot, we say that f is a parking function on [n]. Prove that the number of parking functions on [n] is (n + 1)^n-1
There are n parking spots 1, 2,..., n on a one-way street. Cars 1, 2,..., n arrive in this order. Each car i has a favorite parking spot f(i). When a car arrives, it first goes to its favorite spot. If the spot is free, the car will take it, if not, it goes to the next spot. Again, if that spot is free, the car will take it, if not, the car goes to the next spot. If a car had to leave even the last spot and did not find the space, then its parking attempt has been unsuccessful. If, at the end of this procedure, all cars have a parking spot, we say that f is a parking function on [n]. Prove that the number of parking functions on [n] is (n + 1)^n-1
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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Discrete Math
![There are n parking spots 1, 2, ..., n on a one-way street. Cars 1, 2, ..., n arrive in this order.
Each car i has a favorite parking spot f(i). When a car arrives, it first goes to its favorite spot. If
the spot is free, the car will take it, if not, it goes to the next spot. Again, if that spot is free, the
car will take it, if not, the car goes to the next spot. If a car had to leave even the last spot and
did not find the space, then its parking attempt has been unsuccessful. If, at the end of this
procedure, all cars have a parking spot, we say that f is a parking function on [n]. Prove that the
number of parking functions on [n] is (n + 1)^n-1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2029095-ed0d-4011-975c-f2065b45a57d%2F43e09b77-e062-459b-93db-e8416bfc2303%2Fvsbuyy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:There are n parking spots 1, 2, ..., n on a one-way street. Cars 1, 2, ..., n arrive in this order.
Each car i has a favorite parking spot f(i). When a car arrives, it first goes to its favorite spot. If
the spot is free, the car will take it, if not, it goes to the next spot. Again, if that spot is free, the
car will take it, if not, the car goes to the next spot. If a car had to leave even the last spot and
did not find the space, then its parking attempt has been unsuccessful. If, at the end of this
procedure, all cars have a parking spot, we say that f is a parking function on [n]. Prove that the
number of parking functions on [n] is (n + 1)^n-1
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