The Triplelabeled Algorithm in Parking Functions

1
The Triple-labeled Algorithm in Parking Functions

• Sen-Peng Eu ???
• University of Kaohsiung, Taiwan
• Chun-Ju Lai ???
• National Taiwan University, Taiwan

2
Parking Function (rule)

• There are n parking spots labeled from 1 to n
  along a one-way street. n drivers try to park
  their cars in turn each with one single favorite
  spot in mind.
• The ith driver proceeds to his favorite spot ai.
• If its vacant, then hell park there.
• If not, hell park at the next free spot.
• If no free spots left, then hell give up
  parking.

(a1, a2,..., an)
3
Parking Functions (P3)

• A sequence (a1,...an) is a parking function if
  its nondecreasing rearrangement b1 ... bn
  satisfies bi i for all i
• For example
• 131, 4311, 553112 are parking functions
• 2, 133, 35123, are not parking functions

4
Parking Functions (P3)

• n 1, there are 1 parking functions
• 1
• n 2, there are 3 parking functions
• 11, 12, 21
• n 1, there are 16 parking functions
• 111, 112, 113, 121, 122, 123, 131, 132
• 211, 212, 213, 221, 231, 311, 312, 321

5
Parking Functions (definition of Pnk)

• Let Pn denote the number of ways that n drivers
  park successfully.
• P1 1, P2 3, P3 16, P4 125, ...
• Theorem Konheim Weiss, 1966 There are
  (n1)n-1 parking functions of length n.

6
Parking Functions (definition of Pnk)

• A parking function (a1, ..., an) is k-leading if
  a1 k.
• Let Pn,k denote the number of k-leading parking
  function of length n.

7
Parking Functions (P31)

• The table lists all 16 parking functions of
  length 3.

8
Parking Functions (P31)

• The table lists all 16 parking functions of
  length 3.

P3, 1 8
9
Parking Functions (P32)

• The table lists all 16 parking functions of
  length 3.

P3, 1 8
P3, 2 5
10
Parking Functions (P33)

• The table lists all 16 parking functions of
  length 3.

P3, 1 8
P3, 2 5
P3, 3 3
11
Parking Functions (definition of Pnk)

• Pn,k ?
• Well give an answer by combinatorial argument,
  then move on to prove more.

12
Rooted Labeled Tree (definition of Pnk)

• (n1)n-1 also counts the number of rooted labeled
  trees on the vertex set 0,1, ... , n (rooted at
  0).
• n 3, we have 16 trees.

• Some bijections between trees and parking
  functions are known, but none seems useful.

13
Algorithm (Idea) (

• Given a labeled tree, we label each node again
  according to the Breadth First Search (BFS) and
  let pa(x) denote the 2nd label of node x.

• We then define 3rd label w by the formula
• w(x) pa(parent of x) 1

w(2) 1 w(3) 1 w(6) 1 w(1) 3 w(5)
3 w(4) 4
pa(0) 0
It is proved that (w(1), ...,w(n)) is the
desired parking function. (a1,...an) In this
case, it is (3, 1, 1, 4, 3, 1).
0
pa(2) 1
pa(3) 2 pa(6) 3 pa(1) 4 pa(5) 5 pa(4) 6
1
1
1
1
2
3
3
3
4
4
5
6
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Algorithm (counting)

• Given a parking function (a1,...an) for i 1 to
  n, define
• pa(i) Cardaj?a either aj lt ai or aj ai and
  j lt i
• The triplet-labeled rooted tree Ta associated
  with
• V(Ta) (0, 0, 0) ? (i, ai, pa(i)) aj?a
• Let Ta be rooted at (0, 0, 0)
• For any 2 vertices u (i, ai pa(i)) and v
  (j, aj pa(j)), u is a child of v if ai pa(j)
  1.

15
Enumeration (counting)

• Our algorithm can manipulate on tree and obtain
  Pn,k - Pn,k1 in a neat autograft method
  described below.

16
Enumeration

• Let Tn,k denote the set of all labeled trees
  correspond to k-leading parking functions of
  length n.
• We are to establish a bijection between Tn,k1
  and a subset Tn,k of Tn,k so that Tn,k \ Tn,k
  is easy to compute.

17
Autografting Method
0

• We now perform the "autograft"
• to members of Rn,k as follows

n 5, k 1
1
1
3
1) Identify and remove the subtree S consisting
of node 1 and all its descendants.
3
2
4
2
5

• 2) Renew the labels of the
• remaining nodes according to the BFS.

3) Locate the node y satisfies f(y) k and
re-attach S making node 1 a child of node y.
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Enumeration

• The trees in Tn,k \ Tn,k must be like this

19
Enumeration

• It is easy to observe that
• Tn,k \ Tn,k
• A To choose k-1 points from n-1 points to form
  S.
• B Total ways to form a labeled tree of length k.
• C Total ways to form a labeled tree of length
  n-k1.

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Enumeration (definition of Pnk)

• Theorem

• Theorem

• Theorem

• Theorem

21
Enumeration (definition of Pnk)

• Theorem

Theorem Foata Riordan, 1974 The original
proof is by combining 3 papers.
22
X-Parking Functions (P3)

• Let x (x1,...,xn) be a sequence of positive
  integers. A sequence (a1,...,an) is a x-parking
  function if its nondecreasing rearrangement b1
  ... bn satisfies bi x1 ... xi for all i
• Therefore the ordinary parking function is a
  special case that x (1,1,...,1)
• The enumeration for the generalized parking
  functions is solved.

23
X-Parking Functions (P3)

• An equivalent definition
• Let ? (?1,..., ?n),?1 ... ?n. A sequence
  (a1,...,an) is a ?-parking function its
  nondecreasing rearrangement b1 ... bn
  satisfies bi ?n-i1 for all i
• The ordinary parking function is a special case
  that ? (n, n-1, ...,1)

• Theorem Steck 1968, Gessel 1996.

24
Explicit Formulae

• However, nice explicit formulae are very few.
• Lets back to the x (x1,...,xn) notataion.
• Pitman, Stanley, 1986 (a, b,...,b) and two
  other cases.
• Yan, 1999 Two other cases, algebraically.
• Yan, 2001 (a, b,..., b), combinatorially.
• Kung, Yan, 2001 Goncarov Polynomials.
• Arguably, (a,b,...,b)-parking functions is the
  best so far.

25
Explicit Formulae

• However, nice explicit formulae are very few.
• Lets back to the x (x1,...,xn) notataion.
• How about the Statistics k-leading?
• Foata, Riordan, 1974 (1,1,...,1),
  algebraically.
• Eu, Fu, Lai, 2005, 2005 (a, b,..., b),
  combinatorially.

• No other results

26
K-leading (a,1,...1) Parking Functions

• Consider a forest with a components
• Ex a 2, (2, 5, 9, 1, 5, 7, 2, 4, 1)

(?0 , , )
(?1, , )
0
1
(1, , )
4
(7, , )
2
5
2
(9, , )
3
1
(4, , )
2
1
(5, , )
8
(8, , )
6
4
5
(2, , )
7
5
(6, , )
9
7
(3, , )
9
10
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K-leading (a,1,...1) Parking Functions
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K-leading (a,b,...b) Parking Functions

• When it comes to (a, b, ...,b)-parking functions.
• Consider a forest with a components and
  edge-coloring.
• We extract an (a, 1, ...,1)-parking function.
• Remainder indicates the color used.

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K-leading (a,1,...1) Parking Functions

• Ex a 2, b2, (2, 7, 15, 1, 8, 12, 2, 5, 1)
• r (-1, 1, 1, -1, 0, 0, -1, 1, -1)

(?0 , , )
(?1, , )
0
1
(1, , )
4
(7, , )
2
5
2
(9, , )
3
1
(4, , )
2
1
(5, , )
8
(8, , )
6
8
4
5
5
(2, , )
7
7
5
(6, , )
9
12
7
(3, , )
9
10
15
30
K-leading (a,1,...1) Parking Functions

• Ex a 2, b2, (2, 7, 15, 1, 8, 12, 2, 5, 1)
• r (-1, 1, 1, -1, 0, 0, -1, 1, -1)

(?0 , , )
(?1, , )
0
1
(1, , )
4
2
(7, , )
2
5
(9, , )
3
1
(4, , )
2
1
(5, , )
8
8
(8, , )
6
5
( , , )
(2, , )
7
7
6
6
( , , )
(3, , )
10
15
11
(6, , )
9
( , , )
10
9
12
16
31
K-leading (a,b,...b) Parking Functions
32
Inflating Parking Functions

• Take x (1,1,...,1,a,1, ...,1) of length n,
  where a is at the k-th position. We call it an
  inflating parking function.
• inflating parking functions with a at the k-th
  position ordinary parking functions of length
  na-1 with the first a-1 numbers are ks !

33
Inflating Parking Functions

• Ex x (1, 1, 1, 3, 1, 1, 1, 1, 1, 1)
• From (5, 1, 4, 5, 1, 10, 3, 3, 7)
• to (4, 4, 6, 1, 4, 6, 1, 11, 3, 3, 5)

34
Parking Function (table of Pnk)

• The first few Pn, ks are

1
1
2
8
3
5
50
34
25
16
432
307
243
189
432
35
More Theorems

• Pn,1 Pn,k Pn,n-k2

0
0
0
0
A
1
1
B
A
1
B
1
A
B
B
A
36
More Theorems

• The first few Pn, ks are

1
0
1
1
0
1
1
2
0
8
3
3
3
2
5
0
9
50
34
25
16
9
16
16
432
307
243
189
432
0
125
125
64
64
54
0
1296 625 480
480 625 1296
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More Theorems

• The first few Pn, ks are
• The table is symmetric!

38
More Theorems

• Theorem Eu, Fu Lai, 2005
• Pn,k Pn,k1 Pn,n-k1 Pn,n-k2

39
More Theorems
Pn,bk Pn,bk1
Pn,n-bka Pn,n-bka1
40
Algorithm (counting)
w(x) pa(parent of x) 1
Parking function x-parking function
Tree(forest)
Some ordering
G-parking function
Graph
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• Thank you for your listening