PARKING
FUNCTIONS
Richard P. Stanley
Department of Mathematics
M.I.T. 2-375
Cambridge, MA 02139
rstan@math.mit.edu
http://www-math.mit.edu/~rstan
Transparencies available at:
http://www-math.mit.edu/~rstan/trans.html
1
ENUMERATION OF
PARKING FUNCTIONS
...
n
...
a1
2
a2
an
1
Car Ci prefers space ai. If ai is occupied, then Ci takes the next available
space. We call (a ; : : : ; an) a parking
function (of length n) if all cars can
park.
1
n = 2 : 11 12 21
n = 3 : 111 112 121 211 113 131 311 122
212 221 123 132 213 231 312 321
2
Easy: Let = (a ; : : : ; an) 2 P n .
Let b b bn be the increas1
1
2
ing rearrangement of . Then is a
parking function if and only bi i.
Corollary. Every permutation of
the entries of a parking function is
also a parking function.
3
Theorem (Pyke, 1959; Konheim and
Weiss, 1966). Let f (n) be the number of parking functions of length n.
Then f (n) = (n + 1)n .
Proof (Pollak, c. 1974). Add an additional space n + 1, and arrange the
spaces in a circle. Allow n + 1 also as a
preferred space.
1
...
a1
1
n+1
n
2
3
4
a2
an
Now all cars can park, and there will
be one empty space. is a parking
function if and only if the empty space
is n + 1. If = (a ; : : : ; an) leads to
car Ci parking at space pi, then (a +
j; : : : ; an + j ) (modulo n + 1) will lead
to car Ci parking at space pi + j . Hence
exactly one of the vectors
(a + i; a + i; : : : an + i) (modulo n + 1)
is a parking function, so
(n + 1)n
f (n) = n + 1 = (n + 1)n :
1
1
1
2
1
5
FOREST INVERSIONS
Let F be a rooted forest on the vertex
set f1; : : : ; ng.
5
12
4
2
7
3
10
11
1
9
8
6
THEOREM (Sylvester-Borchardt-
Cayley). The number of such forests
is (n + 1)n .
1
6
1
2
1
3
1
2
2
3
2
3
2
1
3
2
3
1
2
3
1
3
1
2
3
2
1
3
3
1
1
2
1
1
2
2
3
2
3
1
3
1
2
3
2
3
1
2
1
An inversion in F is a pair (i; j ) so
that i > j and i lies on the path from
j to the root.
7
3
inv(F ) = #(inversions of F )
5
12
4
2
7
3
10
11
1
9
8
6
Inversions: (5; 4); (5; 2); (12; 4); (12; 8)
(3; 1); (10; 1); (10; 6); (10; 9)
inv(F ) = 8
8
Let
In ( q ) =
X
F
q
F );
inv(
summed over all forests F with vertex
set f1; : : : ; ng. E.g.,
I (q ) = 1
I (q ) = 2 + q
I (q) = 6 + 6q + 3q + q
Theorem (Mallows-Riordan 1968, GesselWang 1979) We have
X
In(1 + q) = qe G n;
1
2
2
3
3
(
G
)
where G ranges over all connected graphs
(without loops or multiple edges) on
n +1 labelled vertices, and where e(G)
denotes the number of edges of G.
9
Corollary.
P
( )x
n
X
q
x
In(q)(q 1)n n! = Pn ( ) x n
n
n q n
n+1
n
2
0
!
n
0
X
n1
2
0
In(q)(q
1)n
1
n
!
xn = log X q( ) xn
n!
n
!
n
n
2
0
Theorem (Kreweras, 1980) We have
q( )In(1=q) =
X
n
qa 1+
2
a1;:::;a
(
n)
+
a
n
;
where (a ; : : : ; an) ranges over all parking functions of length n.
1
10
NONCROSSING
PARTITIONS
A noncrossing partition of f1; 2; : : : ; ng
is a partition fB ; : : : ; Bk g of f1; : : : ; ng
such that
a < b < c < d; a; c 2 Bi; b; d 2 Bj ) i = j:
1
12
1
11
2
3
10
4
9
5
8
6
7
11
Theorem (H. W. Becker, 1948{49)
The number of noncrossing partitions
of f1; : : : ; ng is the Catalan num-
ber
1 2n
Cn = n + 1 n :
12
A maximal chain m of noncrossing
partitions of f1; : : : ; n+1g is a sequence
; ; ; : : : ; n
0
1
2
of noncrossing partitions of f1; : : : ; n +
1g such that i is obtained from i
by merging two blocks into one. (Hence
i has exactly n + 1 i blocks.)
1 2 3 4 5 1 25 3 4 1 25 34
125 34 12345
1
13
Dene:
min B = least element of B
j < B : j < k 8k 2 B:
Suppose i is obtained from i by
merging together blocks B and B 0, with
min B < min B 0. Dene
i(m) = maxfj 2 B : j < B 0g
(m) = ( (m); : : : ; n(m)):
1
1
For above example:
1 2 3 4 5 1 25 3 4 1 25 34
125 34 12345
we have
(m) = (2; 3; 1; 2):
14
Theorem. is a bijection between
the maximal chains of noncrossing partitions of f1; : : : ; n + 1g and parking
functions of length n.
Corollary (Kreweras, 1972) The number of maximal chains of noncrossing
partitions of f1; : : : ; n + 1g is
(n + 1)n :
1
Is there a connection with Voiculescu's
theory of free probability?
15
THE SHI ARRANGEMENT
Braid arrangement Bn: the set of
hyperplanes
xi xj = 0; 1 i < j n;
in R n .
R = set of regions of Bn
#R = n!
Let R be the \base region"
R : x > x > > xn :
Let d(R) be the number of hyperplanes
in Bn separating R from R.
0
0
1
2
0
16
1
0
2
3
1
x1 =x2
2
x2 =x3
x1 =x3
B3
17
Proposition.
#fR : d(R) = j g =
n
j
g
;
#fw 2 Sn : `(w) =
2
where
`(w) = #f(r; s) : r < s; w(r) > w(s)g;
the number of inversions of w.
X
R2R
qd R =
(
)
(1+ q)(1+ q + q ) (1+ q + + qn ):
2
1
18
Label R with
(R ) = (1; 1; : : : ; 1) 2 Z n :
If R is labelled, R0 is separated from
R only by xi xj = 0 (i < j ), and R0
is unlabelled, then set
(R0) = (R) + ei;
where ei = ith unit coordinate vector.
0
0
R
λ(R)
R’
λ(R’)=λ(R)+ e i
xi = xj
i<j
19
121
221
111
x1 =x2
321
211
311
x1 =x3
x2 =x3
B3
Theorem (easy). The labels of Bn
are the sequences (a ; : : : ; an) 2 Z n
such that 1 ai n i + 1.
1
20
Shi arrangement Sn: the set of
hyperplanes
xi xj = 0; 1; 1 i < j n; in R n :
x2-x 3 =1
x2-x 3 =0
x1-x 2 =0
x1-x 2 =1
x1-x 3=1
21
x1-x 3 =0
base region
R : xn + 1 > x > > xn
0
1
(R ) = (1; 1; : : : ; 1) 2 Z n
If R is labelled, R0 is separated from
0
R only by xi xj = 0 (i < j ), and
R0 is unlabelled, then set
(R0) = (R) + ei:
If R is labelled, R0 is separated from
R only by xi xj = 1 (i < j ), and
R0 is unlabelled, then set
(R0) = (R) + ej :
22
R
λ(R)
R’
R
λ(R’)=λ(R)+e i
λ(R)
xi = xj
i<j
R’
λ(R’)=λ(R)+ej
x i = x j +1
i<j
23
x2-x 3 =1
x2-x 3 =0
312
311
212
213
321
211
x1-x 2 =0
111
113
121
112
123
122
131
221
x1-x 2 =1
231
132
x1-x 3=1
x1-x 3=0
Theorem (Pak, S.). The labels of
Sn are the parking functions of length
n (each occurring once).
24
Corollary (Shi, 1986)
r(Sn) = (n + 1)n
As for Bn, let d(R) be the number
of hyperplanes in Sn separating R and
1
0
R.
Note: If (R) = (a ; : : : ; an), then
d(R) = a + + an n:
Let R = set of regions of Sn.
Corollary.
1
1
X
R2R
qd(R)
= q( )In(1=q):
n
2
25
THE PARKING FUNCTION
Sn-MODULE
The symmetric group acts on the set
Pn of all parking functions of length n
by permuting coordinates.
Sample properties:
Multiplicity of trivial representation
n
(number of orbits) = Cn = n n
n = 3 : 111 211 221 311 321
Number of elements of Pn xed by
w 2 Sn (character value at w):
w
#Fix(w) = (n + 1)
1
2
+1
(# cycles of
26
)
1
For symmetric function acionados: Let PFn = ch(Pn).
PFn =
X
(n + 1)` (
`n
X
)
1
z p 1
1
n )s
s
(1
=
n
+
1
`n
+1
3
2
Y +n
i
5 m
4
1
=
n
n
+
1
i
`n
X n(n 1) (n `() + 2)
=
h
:
m
(
)!
m
(
)!
n
`n
X
1
!PFn =
3
2
Y n+1
X 1
5 m:
4
n
+
1
i
i
`n
27
Moreover
X
PFn tn = (tE ( t))h i ;
n
P
where E (t) = n entn, and h i denotes compositional inverse.
1
+1
0
1
0
28
THE HAIMAN MODULE
The group Sn acts on R = C [x ; : : : ; xn]
by permuting variables, i.e., w xi =
xw i . Let
RSn = ff 2 R : wf = f for all w 2 Sng:
Well-known:
RS = C [e ; : : : ; en];
where
X
ek =
xi xi xi :
i <i <<i n
Let
D = R= RS = R=(e ; : : : ; en):
Then dim D = n!, and Sn acts on D
according to the regular representation.
1
( )
n
1
1
1
1
2
k
n
1
+
29
2
k
Now let Sn act diagonally on
R = C [x ; : : : ; xn; y ; : : : ; yn];
i.e,
w xi = xw i ; w yi = yw i :
As before, let
RSn = ff 2 R : w f = f for all w 2 Sng
D = R= RS :
1
1
( )
( )
n
+
Conjecture (Haiman, 1994). dim D =
(n + 1)n , and the action of Sn on
D is isomorphic to the action on Pn,
tensored with the sign representation.
(Connections with Macdonald polynomials, Hilbert scheme of points in the
plane, etc.)
1
30
A GENERALIZATION
Let
= ( ; : : : ; n); n > 0:
A -parking function is a sequence
(a ; : : : ; an) 2 P n whose increasing rearrangement b bn satises
bi n i .
Ordinary parking functions:
= (n; n 1; : : : ; 1)
1
1
1
1
+1
Number (Steck 1968, Gessel 1996):
2
jn
i+1
i+1 5
N() = n! det 4(j i + 1)!
31
3n
i;j =1
The Parking Function Polytope
(with J. Pitman)
Given x ; : : : ; xn 2 R , dene P =
P (x1; : : : ; xn) R n by: (y ; : : : ; yn) 2
Pn if
0 yi; y + + yi x + + xi
for 1 i n.
Theorem. (a) Let x ; : : : ; xn 2 N .
Then
n! V (Pn) = N ();
where n i = x + + xi.
1
0
1
1
1
1
+1
1
(b) n! V (Pn) =
X
xi xi .
1
parking functions
(i1 ;:::;in )
Note. If each xi > 0, then Pn has
the combinatorial type of an n-cube.
32
n
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 noncrossing partitions, Discrete Math. 126 (1994), 107{119.
4. D. Foata and J. Riordan, Mappings of acyclic and parking
functions, aequationes math. 10 (1974), 10{22.
5. J. Francon, Acyclic and parking functions, J. Combinatorial Theory (A) 18 (1975), 27{35.
6. I. Gessel and D.-L. Wang, Depth-rst search as a combinatorial correspondence, J. Combinatorial Theory (A) 26
(1979), 308{313.
7. M. Haiman, Conjectures on the quotient ring by diagonal
invariants, J. Algebraic Combinatorics 3 (1994), 17{76.
8. P. Headley, Reduced expressions in innite Coxeter groups,
Ph.D. thesis, University of Michigan, Ann Arbor, 1994.
9. P. Headley, On reduced expressions in ane Weyl groups,
in Formal 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 and
applications, SIAM J. Applied Math. 14 (1966), 1266{1274.
11. G. Kreweras, Sur les partitions non croisees d'un cycle, Discrete Math. 1 (1972), 333{350.
12. G. Kreweras, Une famille de polyn^omes ayant plusieurs
proprietes enumeratives, Periodica Math. Hung. 11 (1980),
309{320.
33
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
for labeled trees, Bull Amer. Math. Soc. 74 (1968), 92{94.
15. Y. Poupard, E tude et denombrement paralleles des partitions non croisees d'un cycle et des decoupage d'un polygone convexe, Discrete Math. 2 (1972), 279{288.
16. R. Pyke, The supremum and inmum of the Poisson process, Ann. Math. Statist. 30 (1959), 568{576.
17. V. Reiner, Non-crossing partitions for classical reection
groups, Discrete Math. 177 (1997), 195{222.
18. J.-Y. Shi, The Kazhdan-Lusztig cells in certain ane Weyl
groups, Lecture Note in Mathematics, no. 1179, Springer,
Berlin/Heidelberg/New York, 1986.
19. J.-Y. Shi, Sign types corresponding to an ane Weyl group,
J. London Math. Soc. 35 (1987), 56{74.
20. R. Simion, Combinatorial statistics on non-crossing partitions, J. Combinatorial Theory (A) 66 (1994), 270{301.
21. R. Simion and D. Ullman, On the structure of the lattice of
noncrossing partitions, Discrete Math. 98 (1991), 193{206.
22. R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 298
(1994), 611{624.
23. R. Stanley, Hyperplane arrangements, interval orders, and
trees, Proc. Nat. Acad. Sci. 93 (1996), 2620{2625.
24. R. Stanley, Parking functions and noncrossing partitions,
Electronic J. Combinatorics 4, R20 (1997), 14 pp.
25. R. Stanley, Hyperplane arrangements, parking functions,
and tree inversions, in Mathematical Essays in Honor of
34
Gian-Carlo Rota (B. Sagan and R. Stanley, eds.), Birkhauser,
Boston/Basel/Berlin, 1998, pp. 359{375.
26. G. P. Steck, The Smirnov two-sample tests as rank tests,
Ann. Math. Statist. 40 (1968), 1449-1466.
27. C. H. Yan, Generalized tree inversions and k-parking functions, J. Combinatorial Theory (A) 79 (1997), 268{280.
35