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An activity for high-school math teachers at the MEAD 2010.
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Preferred Parking Perplexity
Virginia BohmeJi Li
The University of Arizona
Tucson High School Teachers’ CircleMEAD
January 30, 2010
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3 2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3
2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 1
3
2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 13
2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 13 2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 13 2
Mission Accomplished!
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4
1
3 2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4 13 2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4 13 2
A piece of cake!
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3 3
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3
3
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 1
3
3
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 13
3
Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 13
3
No where to go!
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Parking function We call a sequence a parking function if all cars end up finding a parking spot.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Parking function We call a sequence a parking function if all cars end up finding a parking spot.
(3, 1, 3, 2) is a parking function.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Parking function We call a sequence a parking function if all cars end up finding a parking spot.
(3, 1, 3, 2) is a parking function.
(4, 1, 3, 2) is a parking function.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Parking function We call a sequence a parking function if all cars end up finding a parking spot.
(3, 1, 3, 2) is a parking function.
(4, 1, 3, 2) is a parking function.
(3, 1, 3, 3) is not a parking function.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
... ... ...
n 3 2 1
a1 a2 ... an
Preferred Parking Perplexity Tucson HS Teacher’s Circle
... ... ...
n 3 2 1
a1 a2 ... an
Each ai is a number from 1 to n
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2
(1)
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2
(1)
(1,2) (2,1) (1,1)
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2
(1)
(1,2) (2,1) (1,1)
List all parking functions of length 3
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2
(1)
(1,2) (2,1) (1,1)
List all parking functions of length 3
(1,1,1) (1,1,2) (1,1,3) ......
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 2 1
3 11
3 2 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3
3 2 1
3 11
3 2 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 1
3 2 1
3 11
3 2 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 11
3 2 1
3 11
3 2 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 11
3 2 1
3 11
3 2 1
1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 11
3 2 1
3 11
3 2 1
3 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1
3 1 1
3 2 1
3 11
3 2 1
3 11
3 2 1
3 1 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
If (a1,a2,...,an) is a parking function, then any permutation of (a1,a2,...,an) is also a parking function.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that does not workSequence that works
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that does not workSequence that works
(1,1,1)
(1,1,2) (1,2,1) (2,1,1)
(1,1,3) (1,3,1) (3,1,1)
(1,2,2) (2,1,2) (2,2,1)
(1,2,3)(1,3,2)(2,1,3)(2,3,1)(3,1,2)(3,2,1)
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that does not workSequence that works
(1,1,1)
(1,1,2) (1,2,1) (2,1,1)
(1,1,3) (1,3,1) (3,1,1)
(1,2,2) (2,1,2) (2,2,1)
(1,2,3)(1,3,2)(2,1,3)(2,3,1)(3,1,2)(3,2,1)
(1,1,1)
(1,1,2) (1,2,1) (2,1,1)
(1,1,3) (1,3,1) (3,1,1)
(1,2,2) (2,1,2) (2,2,1)
(1,2,3)(1,3,2)(2,1,3)(2,3,1)(3,1,2)(3,2,1)
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Preferred Parking Perplexity Tucson HS Teacher’s Circle
A multiset is such
that each member
has a multiplicity
indicating how many
times it is a member.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
A multiset is such
that each member
has a multiplicity
indicating how many
times it is a member.
For example, in the multiset {a,a,b,c}, the multiplicities of a, b, and c are 2, 1, and 1.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
A multiset is such
that each member
has a multiplicity
indicating how many
times it is a member.
For example, in the multiset {a,a,b,c}, the multiplicities of a, b, and c are 2, 1, and 1.
How many permutations of {a, a, b, c} are there?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)
1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)
1444
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
1444
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
14446
1212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)
14446
1212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)
1 6124
446
1212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)(1,2,2,2)(1,2,2,3)(1,2,2,4)
1 6124
446
1212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)(1,2,2,2)(1,2,2,3)(1,2,2,4)
1 6124
446
1212
41212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)(1,2,2,2)(1,2,2,3)(1,2,2,4)(1,2,3,3)(1,2,3,4)
1 6124
446
1212
41212
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)(1,2,2,2)(1,2,2,3)(1,2,2,4)(1,2,3,3)(1,2,3,4)
1 6124
446
1212
412121224
Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)(1,1,1,2)(1,1,1,3)(1,1,1,4)(1,1,2,2)(1,1,2,3)(1,1,2,4)
(1,1,3,3)(1,1,3,4)(1,2,2,2)(1,2,2,3)(1,2,2,4)(1,2,3,3)(1,2,3,4)
1 6124
446
1212
412121224
125
Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?
The number of sequences (a1,a2,...,an) where each ai is a number from 1 to n is
Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?
The number of sequences (a1,a2,...,an) where each ai is a number from 1 to n is nn
Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?
Let f(n) be the number of parking functions.
The number of sequences (a1,a2,...,an) where each ai is a number from 1 to n is nn
Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?
Let f(n) be the number of parking functions. n 1 2 3 4
f(n) 1 3 16 125nn 1 4 27 256
The number of sequences (a1,a2,...,an) where each ai is a number from 1 to n is nn
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2
...an
a1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2
...an
a1
Each ai is a number from 1 to n+1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
6
66
6
6
6
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
6
6
6
6
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
6
6
6
6
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
66
6
6n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
66
6
6
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
6
6
6
6
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
6
6
6
6
Done!n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
1
2
3
4
5
66
6
6
6
6
Done!
A sequence (a1, ... an) is called a circular parking function if all cars can be parked.
n=5
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Q1: How many circular parking functions are there?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many circular parking functions are there?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many circular parking functions are there?
Any sequence (a1, a2, ..., an) where each ai is between 1 and n+1 gives a circular parking function.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many circular parking functions are there?
Any sequence (a1, a2, ..., an) where each ai is between 1 and n+1 gives a circular parking function.
There are (n+1)n such.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Q2: What happens if spot n+1 is empty?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Q2: What happens if spot n+1 is empty?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Preferred Parking Perplexity Tucson HS Teacher’s Circle
a2...
an
Each ai is a number from 1 to n+1
a1
Q3: Which spot is most likely to be left empty?
Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
The probability that any particular spot is left empty is 1/(n+1).
There are (n+1)n circular parking functions.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
The probability that any particular spot is left empty is 1/(n+1).
There are (n+1)n-1 circular parking functions leaving spot n+1 empty.
There are (n+1)n circular parking functions.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
The probability that any particular spot is left empty is 1/(n+1).
There are (n+1)n-1 circular parking functions leaving spot n+1 empty.
There are (n+1)n-1 parking functions on n.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n-1 parking functions on n.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Let f(n) be the number of parking functions.
There are (n+1)n-1 parking functions on n.
Preferred Parking Perplexity Tucson HS Teacher’s Circle
Let f(n) be the number of parking functions.
n 1 2 3 4 5 6 7 8
f(n) 1 3 16 125 1296 16807 262144 4782969
There are (n+1)n-1 parking functions on n.
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