The document discusses parking functions, which are sequences where cars are assigned parking spots without any car being unable to find a spot. It provides examples of parking functions and non-parking functions of different lengths. It also notes that any permutation of a parking function sequence is also a parking function.

1. Preferred Parking Perplexity
Virginia Bohme
Ji Li
The University of Arizona
Tucson High School Teachers’ Circle
MEAD
January 30, 2010

2. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3 2

3. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3
2

4. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 1
3
2

5. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 3 1
2

6. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 3 2 1

7. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 3 2 1
Mission Accomplished!

8. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4
1
3 2

9. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4 3 2 1

10. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
4 3 2 1
A piece of cake!

11. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3 3

12. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3
1
3
3

13. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 1
3
3

14. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 3 1
3

15. Preferred Parking Perplexity Tucson HS Teacher’s Circle
4 3 2 1
3 3 1
3
No where to go!

16. Preferred Parking Perplexity Tucson HS Teacher’s Circle
We call a sequence a parking
Parking function function if all cars end up
finding a parking spot.

17. Preferred Parking Perplexity Tucson HS Teacher’s Circle
We call a sequence a parking
Parking function function if all cars end up
finding a parking spot.
(3, 1, 3, 2) is a parking function.

18. Preferred Parking Perplexity Tucson HS Teacher’s Circle
We call a sequence a parking
Parking function 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.

19. Preferred Parking Perplexity Tucson HS Teacher’s Circle
We call a sequence a parking
Parking function 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.

20. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n 3 2 1
... ... ...
a1 a2 ... an

21. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n 3 2 1
... ... ...
a1 a2 ... an
Each ai is a number from 1 to n

22. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
List all parking functions of length 2

23. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
(1)
List all parking functions of length 2

24. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
(1)
List all parking functions of length 2
(1,2) (2,1) (1,1)

25. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
(1)
List all parking functions of length 2
(1,2) (2,1) (1,1)
List all parking functions of length 3

26. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 1
(1)
List all parking functions of length 2
(1,2) (2,1) (1,1)
List all parking functions of length 3
(1,1,1) (1,1,2) (1,1,3) ......

27. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1 1
3 2 1 3 2 1
1 3 1

28. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3
3 1 1
3 2 1 3 2 1
1 3 1

29. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1
3 1 1
3 2 1 3 2 1
1 3 1

30. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1 1
3 1 1
3 2 1 3 2 1
1 3 1

31. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1 1
3 1 1
3 2 1 3 2 1
1
1 3 1

32. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1 1
3 1 1
3 2 1 3 2 1
3 1
1 3 1

33. Preferred Parking Perplexity Tucson HS Teacher’s Circle
3 2 1 3 2 1
3 1 1
3 1 1
3 2 1 3 2 1
3 1 1
1 3 1

34. 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.

35. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that works Sequence that does not work

36. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that works Sequence that does not work
(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)

37. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 3
Sequence that works Sequence that does not work
(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)

38. Preferred Parking Perplexity Tucson HS Teacher’s Circle

39. Preferred Parking Perplexity Tucson HS Teacher’s Circle
e t is s uch
m ul t is be r
A me m
e ach l ic i t y
t h at u l t i p
h as am
how m any
ca t i ng b e r.
i n di a me m
t i me s i t is

40. Preferred Parking Perplexity Tucson HS Teacher’s Circle
e t is s uch
m ul t is be r Fo r e xam
A me m
e ach l ic i t y p le , i n t h
t h at u l t i p {a, a, b,c} e mu l t i s e
, t h e mu t
h as am
how m any o f a, b, a
n d c a re
l t i p lic i t i
es
ca t i ng b e r. 2, 1, a n d
i n di a me m 1.
t i me s i t is

41. Preferred Parking Perplexity Tucson HS Teacher’s Circle
e t is s uch
m ul t is be r Fo r e xam
A me m
e ach l ic i t y p le , i n t h
t h at u l t i p {a, a, b,c} e mu l t i s e
, t h e mu t
h as am
how m any o f a, b, a
n d c a re
l t i p lic i t i
es
ca t i ng b e r. 2, 1, a n d
i n di a me m 1.
t i me s i t is
How many permutations of {a, a, b, c} are there?

42. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1)

43. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1

44. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1
(1,1,1,2)
(1,1,1,3)
(1,1,1,4)

45. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1
(1,1,1,2) 4
(1,1,1,3) 4
(1,1,1,4) 4

46. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1
(1,1,1,2) 4
(1,1,1,3) 4
(1,1,1,4) 4
(1,1,2,2)
(1,1,2,3)
(1,1,2,4)

47. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1
(1,1,1,2) 4
(1,1,1,3) 4
(1,1,1,4) 4
(1,1,2,2) 6
(1,1,2,3) 12
(1,1,2,4) 12

48. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3)
(1,1,3,4)
(1,1,1,2) 4
(1,1,1,3) 4
(1,1,1,4) 4
(1,1,2,2) 6
(1,1,2,3) 12
(1,1,2,4) 12

49. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4
(1,1,1,4) 4
(1,1,2,2) 6
(1,1,2,3) 12
(1,1,2,4) 12

50. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4 (1,2,2,2)
(1,1,1,4) 4 (1,2,2,3)
(1,2,2,4)
(1,1,2,2) 6
(1,1,2,3) 12
(1,1,2,4) 12

51. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4 (1,2,2,2) 4
(1,1,1,4) 4 (1,2,2,3) 12
(1,2,2,4) 12
(1,1,2,2) 6
(1,1,2,3) 12
(1,1,2,4) 12

52. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4 (1,2,2,2) 4
(1,1,1,4) 4 (1,2,2,3) 12
(1,2,2,4) 12
(1,1,2,2) 6
(1,2,3,3)
(1,1,2,3) 12
(1,2,3,4)
(1,1,2,4) 12

53. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4 (1,2,2,2) 4
(1,1,1,4) 4 (1,2,2,3) 12
(1,2,2,4) 12
(1,1,2,2) 6
(1,2,3,3) 12
(1,1,2,3) 12
(1,2,3,4) 24
(1,1,2,4) 12

54. Preferred Parking Perplexity Tucson HS Teacher’s Circle
List all parking functions of length 4 125
(1,1,1,1) 1 (1,1,3,3) 6
(1,1,3,4) 12
(1,1,1,2) 4
(1,1,1,3) 4 (1,2,2,2) 4
(1,1,1,4) 4 (1,2,2,3) 12
(1,2,2,4) 12
(1,1,2,2) 6
(1,2,3,3) 12
(1,1,2,3) 12
(1,2,3,4) 24
(1,1,2,4) 12

55. Preferred Parking Perplexity Tucson HS Teacher’s Circle
How many parking functions are there for n?

56. 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

57. 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

58. 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
Let f(n) be the number of
parking functions.

59. 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
Let f(n) be the number of
parking functions. n 1 2 3 4
f(n) 1 3 16 125
nn 1 4 27 256

60. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an

61. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1 to n+1

62. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5 6
1
6
6
2 6 6
6
3 5
4

63. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1
6
6
2 6 6
6 6
3 5
4

64. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1
6 6
2 6 6
6 6
3 5
4

65. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1
6
6
6
2 6
6 6
3 5
4

66. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1
6
6
2 6
6 6
3 5
6
4

67. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1
6
2 6
6 6
3 5
6
4
6

68. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1 Done!
6
2 6
6 6
3 5
6
4
6

69. Preferred Parking Perplexity Tucson HS Teacher’s Circle
n=5
1 Done!
6
2 6
A sequence
6 6
(a1, ... an) is called
a circular parking
3 5
function if all
cars can be
6 parked.
4
6

70. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1

71. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1
Q1: How many circular
parking functions are
there?

72. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many
circular parking
functions are
there?

73. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many
circular parking
functions are Any sequence (a1, a2, ..., an)
where each ai is bet ween
there? 1 and n+1 gives a circular
parking function.

74. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1 a2 ... an
How many
circular parking
functions are Any sequence (a1, a2, ..., an)
where each ai is bet ween
there? 1 and n+1 gives a circular
parking function.
There are (n+1)n such.

75. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1

76. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1
Q2: What happens if
spot n+1 is empty?

77. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1
Q2: What happens if
spot n+1 is empty?

78. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1

79. Preferred Parking Perplexity Tucson HS Teacher’s Circle
a1
a2
...
an
Each ai is a number from 1
to n+1
Q3: Which spot is
most likely to be
left empty?

80. Preferred Parking Perplexity Tucson HS Teacher’s Circle

81. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.

82. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.
The probability that any particular spot is left
empty is 1/(n+1).

83. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.
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.

84. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n circular parking functions.
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.

85. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n-1 parking functions on n.

86. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n-1 parking functions on n.
Let f(n) be the number of
parking functions.

87. Preferred Parking Perplexity Tucson HS Teacher’s Circle
There are (n+1)n-1 parking functions on n.
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

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