a survey of parking functionsrstan/transparencies/parking3.pdf · parking functions..... n 2 1 a a...

A Survey of Parking Functions

Richard P. Stanley

M.I.T.

A Survey of Parking Functions – p. 1

Parking functions

...

...

n 2 1a a a1 2 n

Car Ci prefers space ai. If ai is occupied, then Ci

takes the next available space. We call(a1, . . . , an) a parking function (of length n) if allcars can park.

n = 2 : 11 12 21

n = 3 : 111 112 121 211 113 131 311 122

212 221 123 132 213 231 312 321


Permutations of parking functions

Easy: Let α = (a1, . . . , an) ∈ Pn. Let

b1 ≤ b2 ≤ · · · ≤ bn be the increasingrearrangement of α. Then α is a parking functionif and only bi ≤ i.

Corollary. Every permutation of the entries of aparking function is also a parking function.


Enumeration of parking functions

Theorem (Pyke , 1959; Konheim and Weiss ,1966). Let f(n) be the number of parkingfunctions of length n. Then f(n) = (n + 1)n−1.

Proof (Pollak , c. 1974). Add an additional spacen + 1, and arrange the spaces in a circle. Allown + 1 also as a preferred space.


a a a1 2

...n

1

n2

3

n+1


Conclusion of Pollak’s proof

Now all cars can park, and there will be oneempty space. α is a parking function ⇔ if theempty space is n + 1. If α = (a1, . . . , an) leads tocar Ci parking at space pi, then (a1 + j, . . . , an + j)(modulo n + 1) will lead to car Ci parking atspace pi + j. Hence exactly one of the vectors

(a1 + i, a2 + i, . . . , an + i) (modulo n + 1)

is a parking function, so

f(n) =(n + 1)n

n + 1= (n + 1)n−1.


Forest inversions

Let F be a rooted forest on the vertex set{1, . . . , n}.

1

6

9

10

37

112

84

5

12

Theorem (Sylvester-Borchardt-Cayley ). Thenumber of such forests is (n + 1)n−1.


The case n = 3

132

1

3

1

3

2

3

1

2

2

3

1

3

2

1

21

1 2 3 1

2

3 1

32

2

31

2

13

3

12

3

32

2

11

2

3

1

2

3


Forest inversions

An inversion in F is a pair (i, j) so that i > j andi lies on the path from j to the root.

inv(F ) = #(inversions of F )


Example of forest inversions

1

6

9

10

37

112

84

5

12


Example of forest inversions

1

6

9

10

37

112

84

5

12

Inversions: (5, 4), (5, 2), (12, 4), (12, 8)

(3, 1), (10, 1), (10, 6), (10, 9)

inv(F ) = 8


The inversion enumerator

LetIn(q) =

∑

F

qinv(F ),

summed over all forests F with vertex set{1, . . . , n}. E.g.,

I1(q) = 1

I2(q) = 2 + q

I3(q) = 6 + 6q + 3q2 + q3


Relation to connected graphs

Theorem (Mallows-Riordan 1968,Gessel-Wang 1979) We have

In(1 + q) =∑

G

qe(G)−n,

where G ranges over all connected graphs(without loops or multiple edges) on n + 1labelled vertices, and where e(G) denotes thenumber of edges of G.


A generating function

Corollary.

∑

n≥0

In(q)(q − 1)nxn

n!=

∑

n≥0 q(n+1

2 )xn

n!∑

n≥0 q(n

2)xn

n!

∑

n≥1

In(q)(q − 1)n−1xn

n!= log

∑

n≥0

q(n

2)xn

n!


Relation to parking functions

Theorem (Kreweras , 1980). We have

q(n

2)In(1/q) =∑

(a1,...,an)

qa1+···+an,

where (a1, . . . , an) ranges over all parkingfunctions of length n.


Noncrossing partitions

A noncrossing partition of {1, 2, . . . , n} is apartition {B1, . . . , Bk} of {1, . . . , n} such that

a < b < c < d, a, c ∈ Bi, b, d ∈ Bj ⇒ i = j.

12

11

10

9

87 6

5

4

3

2

1


Enumeration of noncrossing partitions

Theorem (H. W. Becker , 1948–49). The numberof noncrossing partitions of {1, . . . , n} is theCatalan number

Cn =1

n + 1

(

2n

n

)

.


Chains of noncrossing partitions

A maximal chain m of noncrossing partitions of{1, . . . , n + 1} is a sequence

π0, π1, π2, . . . , πn

of noncrossing partitions of {1, . . . , n + 1} suchthat πi is obtained from πi−1 by merging twoblocks into one. (Hence πi has exactly n + 1 − iblocks.)


Chains of noncrossing partitions

A maximal chain m of noncrossing partitions of{1, . . . , n + 1} is a sequence

π0, π1, π2, . . . , πn

of noncrossing partitions of {1, . . . , n + 1} suchthat πi is obtained from πi−1 by merging twoblocks into one. (Hence πi has exactly n + 1 − iblocks.)

1−2−3−4−5 1−25−3−4 1−25−34

125−34 12345


A chain labeling

Define:

min B = least element of B

j < B : j < k ∀k ∈ B.

Suppose πi is obtained from πi−1 by mergingtogether blocks B and B′, with min B < min B′.Define

Λi(m) = max{j ∈ B : j < B′}

Λ(m) = (Λ1(m), . . . ,Λn(m)).


An example

1−2−3−4−5 1−25−3−4 1−25−34

125−34 12345

we haveΛ(m) = (2, 3, 1, 2).


Number of chains

Theorem. Λ is a bijection between the maximalchains of noncrossing partitions of {1, . . . , n + 1}and parking functions of length n.


Number of chains


Corollary (Kreweras, 1972). The number ofmaximal chains of noncrossing partitions of{1, . . . , n + 1} is (n + 1)n−1.


Number of chains


Corollary (Kreweras, 1972). The number ofmaximal chains of noncrossing partitions of{1, . . . , n + 1} is (n + 1)n−1.

Is there a connection with Voiculescu’s theory offree probability?


The Shi arrangement: background

Braid arrangement Bn: the set of hyperplanes

xi − xj = 0, 1 ≤ i < j ≤ n,

in Rn.




xi − xj = 0, 1 ≤ i < j ≤ n,

in Rn.

R = set of regions of Bn

#R = ??




xi − xj = 0, 1 ≤ i < j ≤ n,

in Rn.


#R = n!




xi − xj = 0, 1 ≤ i < j ≤ n,

in Rn.


#R = n!

Let R0 be the base region

R0 : x1 > x2 > · · · > xn.


Labeling the regions

Label R0 with

λ(R0) = (1, 1, . . . , 1) ∈ Zn.

If R is labelled, R′ is separated from R only byxi − xj = 0 (i < j), and R′ is unlabelled, then set

λ(R′) = λ(R) + ei,

where ei = ith unit coordinate vector.


The labeling rule

λ( )=λ( )+eR iR’λ( )R

x = xi < j

ji

RR’


Description of labels

211311

121

321

111 221

B3x =x1

x =x1

x =x23

3

2


Description of labels

211311

121

321

111 221

B3x =x1

x =x1

x =x23

3

2

Theorem (easy). The labels of Bn are thesequences (b1, . . . , bn) ∈ Z

n such that1 ≤ bi ≤ n − i + 1.


Separating hyperplanes

RecallR0 : x1 > x2 > · · · > xn.

Let d(R) be the number of hyperplanes in Bn

separating R0 from R.


The case n = 3

01

231

2

1

x =xx =x 3

x =x

1

2

2

B3

3


Generating function for d(R)

NOTE: If λ(R) = (b1, . . . , bn), thend(R) =

∑

(bi − 1).


Generating function for d(R)

NOTE: If λ(R) = (b1, . . . , bn), thend(R) =

∑

(bi − 1).

Easy consequence:

Corollary.∑

R∈R

qd(R) =

(1 + q)(1 + q + q2) · · · (1 + q + · · · + qn−1).


The Shi arrangement

Shi Jianyi ( )


The Shi arrangement

Shi Jianyi ( )

Shi arrangement Sn: the set of hyperplanes

xi − xj = 0, 1,

1 ≤ i < j ≤ n, in Rn.


The case n = 3

2

1

1

1 1

x -x =1 x -x =0

x -x =0

x -x =1

x -x =1 x -x =03

2

3 3

3

2

2



base region:

R0 : xn + 1 > x1 > · · · > xn



base region:

R0 : xn + 1 > x1 > · · · > xn

λ(R0) = (1, 1, . . . , 1) ∈ Zn


If R is labelled, R′ is separated from R only byxi − xj = 0 (i < j), and R′ is unlabelled, thenset

λ(R′) = λ(R) + ei.

If R is labelled, R′ is separated from R only byxi − xj = 1 (i < j), and R′ is unlabelled, thenset

λ(R′) = λ(R) + ej.


The labeling rule

λ( )=λ( )+eR iR’λ( )R

x = xi < j

ji

RR’

λ( )R

+1x = xi < j

ji

RR’

λ( )=λ( )+eRR’ j


The labeling for n = 2

123 122132

131 231

121111

112

211

212 311

312

321213

221113

2

1

1

1 1

x −x =1 x −x =0

x −x =0

x −x =1

x −x =1 x −x =0

2

3 3

2

2

3 3


Description of the labels

Theorem (Pak, S.). The labels of Sn are theparking functions of length n (each occurringonce).


Description of the labels

Theorem (Pak, S.). The labels of Sn are theparking functions of length n (each occurringonce).

Corollary (Shi , 1986)

r(Sn) = (n + 1)n−1


The parking function Sn-module

The symmetric group Sn acts on the set Pn ofall parking functions of length n by permutingcoordinates.


Sample properties

Multiplicity of trivial representation (number oforbits) = Cn = 1

n+1

(

2nn

)

n = 3 : 111 211 221 311 321

Number of elements of Pn fixed by w ∈ Sn

(character value at w):

#Fix(w) = (n + 1)(# cycles of w)−1


Symmetric functions

For symmetric function aficionados: LetPFn = ch(Pn).

PFn =∑

λ⊢n

(n + 1)ℓ(λ)−1z−1λ pλ

=∑

λ⊢n

1

n + 1sλ(1

n+1)sλ

=∑

λ⊢n

1

n + 1

[

∏

i

(

λi + n

n

)

]

mλ


More properties

PFn =∑

λ⊢n

n(n − 1) · · · (n − ℓ(λ) + 2)

m1(λ)! · · ·mn(λ)!hλ.

ωPFn =∑

λ⊢n

1

n + 1

[

∏

i

(

n + 1

λi

)

]

mλ.


Background: invariants of Sn

The group Sn acts on R = C[x1, . . . , xn] bypermuting variables, i.e., w · xi = xw(i). Let

RSn = {f ∈ R : w · f = f for all w ∈ Sn}.


Background: invariants of Sn

The group Sn acts on R = C[x1, . . . , xn] bypermuting variables, i.e., w · xi = xw(i). Let

RSn = {f ∈ R : w · f = f for all w ∈ Sn}.

Well-known:

RSn = C[e1, . . . , en],

where

ek =∑

1≤i1<i2<···<ik≤n

xi1xi2 · · · xik.


The coinvariant algebra

LetD = R/

(

RSn

+

)

= R/(e1, . . . , en).

Then dim D = n!, and Sn acts on D according tothe regular representation .


Diagonal action of Sn

Now let Sn act diagonally on

R = C[x1, . . . , xn, y1, . . . , yn],

i.e,w · xi = xw(i), w · yi = yw(i).

As before, let

RSn = {f ∈ R : w · f = f for all w ∈ Sn}

D = R/(

RSn

+

)

.


Hainman’s theorem

Theorem (Haiman , 1994, 2001).dim D = (n + 1)n−1, and the action of Sn on D isisomorphic to the action on Pn, tensored with thesign representation. (Connections withMacdonald polynomials, Hilbert scheme of pointsin the plane, etc.)


λ-parking functions

Let

λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn > 0.

A λ-parking function is a sequence(a1, . . . , an) ∈ P

n whose increasingrearrangement b1 ≤ · · · ≤ bn satisfies bi ≤ λn−i+1.


λ-parking functions

Let

λ = (λ1, . . . , λn), λ1 ≥ · · · ≥ λn > 0.

A λ-parking function is a sequence(a1, . . . , an) ∈ P

n whose increasingrearrangement b1 ≤ · · · ≤ bn satisfies bi ≤ λn−i+1.

Ordinary parking functions:

λ = (n, n − 1, . . . , 1)


Enumeration of λ-parking fns.

Theorem (Steck 1968, Gessel 1996). Thenumber N(λ) of λ-parking functions is given by

N(λ) = n! det

[

λj−i+1n−i+1

(j − i + 1)!

]n

i,j=1


The parking function polytope

Given x1, . . . , xn ∈ R≥0, defineP = P(x1, . . . , xn) ⊂ R

n by: (y1, . . . , yn) ∈ Pn if

0 ≤ yi, y1 + · · · + yi ≤ x1 + · · · + xi

for 1 ≤ i ≤ n.


The parking function polytope

Given x1, . . . , xn ∈ R≥0, defineP = P(x1, . . . , xn) ⊂ R

n by: (y1, . . . , yn) ∈ Pn if

0 ≤ yi, y1 + · · · + yi ≤ x1 + · · · + xi

for 1 ≤ i ≤ n.(also called Pitman-Stanley polytope )


Volume of P

Theorem. (a) Let x1, . . . , xn ∈ N. Then

n!V (Pn) = N(λ),

where λn−i+1 = x1 + · · · + xi.

(b) n!V (Pn) =∑

parking functions(i1,...,in)

xi1 · · · xin.


Volume of P

Theorem. (a) Let x1, . . . , xn ∈ N. Then

n!V (Pn) = N(λ),

where λn−i+1 = x1 + · · · + xi.

(b) n!V (Pn) =∑

parking functions(i1,...,in)

xi1 · · · xin.

NOTE. If each xi > 0, then Pn has thecombinatorial type of an n-cube.


References

1. H. W. Becker, Planar rhyme schemes, Bull. Amer. Math. Soc. 58(1952), 39. Math. Mag. 22 (1948–49), 23–26

2. P. H. Edelman, Chain enumeration and non-crossing partitions,Discrete Math. 31 (1980), 171–180.

3. P. H. Edelman and R. Simion, Chains in the lattice of noncrossingpartitions, Discrete Math. 126 (1994), 107–119.

4. D. Foata and J. Riordan, Mappings of acyclic and parkingfunctions, aequationes math. 10 (1974), 10–22.

5. D. Chebikin and A. Postnikov, Generalized parking functions,

descent numbers, and chain polytopes of ribbon posets,

Advances in Applied Math., to appear.


6. I. Gessel and D.-L. Wang, Depth-first search as a combinatorialcorrespondence, J. Combinatorial Theory (A) 26 (1979), 308–313.

7. M. Haiman, Conjectures on the quotient ring by diagonalinvariants, J. Algebraic Combinatorics 3 (1994), 17–76.

8. P. Headley, Reduced expressions in infinite Coxeter groups, Ph.D.thesis, University of Michigan, Ann Arbor, 1994.

9. P. Headley, On reduced expressions in affine Weyl groups, inFormal Power Series and Algebraic Combinatorics, FPSAC ’94,May 23–27, 1994, DIMACS preprint, pp. 225–232.

10. A. G. Konheim and B. Weiss, An occupancy discipline andapplications, SIAM J. Applied Math. 14 (1966), 1266–1274.


11. G. Kreweras, Sur les partitions non croisées d’un cycle, DiscreteMath. 1 (1972), 333–350.

12. G. Kreweras, Une famille de polynômes ayant plusieurs propriétésénumeratives, Periodica Math. Hung. 11 (1980), 309–320.

13. J. Lewis, Parking functions and regions of the Shi arrangement,preprint dated 1 August 1996.

14. C. L. Mallows and J. Riordan, The inversion enumerator forlabeled trees, Bull Amer. Math. Soc. 74 (1968), 92–94.

15. J. Pitman and R. Stanley, A polytope related to empirical

distributions, plane trees, parking functions, and the

associahedron, Discrete Comput. Geom. 27 (2002), 603–634.


16. A. Postnikov and B. Shapiro„ Trees, parking functions, syzygies,and deformations of monomial ideals, Trans. Amer. Math. Soc.356 (2004), 3109–3142.

17. Y. Poupard, Étude et denombrement paralleles des partitions noncroisées d’un cycle et des decoupage d’un polygone convexe,Discrete Math. 2 (1972), 279–288.

18. R. Pyke, The supremum and infimum of the Poisson process,Ann. Math. Statist. 30 (1959), 568–576.

19. V. Reiner, Non-crossing partitions for classical reflection groups,Discrete Math. 177 (1997), 195–222.

20. J.-Y. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups,

Lecture Note in Mathematics, no. 1179, Springer,

Berlin/Heidelberg/New York, 1986.A Survey of Parking Functions – p. 50

21. J.-Y. Shi, Sign types corresponding to an affine Weyl group, J.London Math. Soc. 35 (1987), 56–74.

22. R. Simion, Combinatorial statistics on non-crossing partitions, J.Combinatorial Theory (A) 66 (1994), 270–301.

23. R. Simion and D. Ullman, On the structure of the lattice ofnoncrossing partitions, Discrete Math. 98 (1991), 193–206.

24. R. Speicher, Multiplicative functions on the lattice of noncrossingpartitions and free convolution, Math. Ann. 298 (1994), 611–624.

25. R. Stanley, Hyperplane arrangements, interval orders, and trees,

Proc. Nat. Acad. Sci. 93 (1996), 2620–2625.


26. R. Stanley, Parking functions and noncrossing partitions,Electronic J. Combinatorics 4, R20 (1997), 14 pp.

27. R. Stanley, Hyperplane arrangements, parking functions, and treeinversions, in Mathematical Essays in Honor of Gian-Carlo Rota(B. Sagan and R. Stanley, eds.), Birkhäuser, Boston/Basel/Berlin,1998, pp. 359–375.

28. G. P. Steck, The Smirnov two-sample tests as rank tests, Ann.Math. Statist. 40 (1969), 1449-1466.

29. C. H. Yan, Generalized tree inversions and k-parking functions, J.

Combinatorial Theory (A) 79 (1997), 268–280.


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