A Survey of Parking Functions

Transcription

1 A Survey of Parking Functions Richard P. Stanley M.I.T.

2 Parking functions... n a a a 1 2 n Car C i prefers space a i. If a i is occupied, then C i takes the next available space. We call (a 1,...,a n ) a parking function (of length n) if all cars can park. n = 2 : n = 3 :

3 Permutations of parking functions Easy: Let α = (a 1,...,a n ) P n. Let b 1 b 2 b n be the increasing rearrangement of α. Then α is a parking function if and only b i i. Corollary. Every permutation of the entries of a parking function is also a parking function.

4 Enumeration of parking functions 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 1. 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.

5 1 n+1 a a a n 2 n 3

6 Conclusion of Pollak s proof Now all cars can park, and there will be one empty space. α is a parking function if the empty space is n + 1. If α = (a 1,...,a n ) leads to car C i parking at space p i, then (a 1 +j,...,a n +j) (modulo n + 1) will lead to car C i parking at space p i + j. Hence exactly one of the vectors (a 1 + i,a 2 + i,...,a n + i) (modulo n + 1) is a parking function, so f(n) = (n + 1)n n + 1 = (n + 1)n 1.

7 Forest inversions Let F be a rooted forest on the vertex set {1,...,n} Theorem (Sylvester-Borchardt-Cayley). The number of such forests is (n + 1) n

8 The case n =

9 Forest inversions An inversion in F is a pair (i,j) so that i > j and i lies on the path from j to the root. inv(f) = #(inversions of F )

10 Example of forest inversions

11 Example of forest inversions Inversions: (5, 4), (5, 2), (12, 4), (12, 8) (3, 1), (10, 1), (10, 6), (10, 9) inv(f) = 8

12 The inversion enumerator Let I n (q) = F q inv(f), summed over all forests F with vertex set {1,...,n}. E.g., I 1 (q) = 1 I 2 (q) = 2 + q I 3 (q) = 6 + 6q + 3q 2 + q 3

13 Relation to connected graphs Theorem (Mallows-Riordan 1968, Gessel-Wang 1979) We have I n (1 + q) = G q e(g) n, 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.

14 A generating function Corollary. n 0 n 1 I n (q)(q 1) nxn n! = n 0 q(n+1 2 ) x n n! 2) x n 0 n q(n n! I n (q)(q 1) n 1xn n! = log n 0 q (n 2) xn n!

15 Relation to parking functions Theorem (Kreweras, 1980). We have q 2) (n In (1/q) = q a 1+ +a n, (a 1,...,a n ) where (a 1,...,a n ) ranges over all parking functions of length n.

16 Noncrossing partitions A noncrossing partition of {1, 2,...,n} is a partition {B 1,...,B k } of {1,...,n} such that a < b < c < d, a,c B i, b,d B j i = j

17 Enumeration of noncrossing partitio Theorem (H. W. Becker, ). The number of noncrossing partitions of {1,...,n} is the Catalan number C n = 1 n + 1 ( 2n n ).

18 Chains of noncrossing partitions A maximal chain m of noncrossing partitions of {1,...,n + 1} is a sequence π 0, π 1, π 2,..., π n of noncrossing partitions of {1,...,n + 1} such that π i is obtained from π i 1 by merging two blocks into one. (Hence π i has exactly n + 1 i blocks.)

19 Chains of noncrossing partitions A maximal chain m of noncrossing partitions of {1,...,n + 1} is a sequence π 0, π 1, π 2,..., π n of noncrossing partitions of {1,...,n + 1} such that π i is obtained from π i 1 by merging two blocks into one. (Hence π i has exactly n + 1 i blocks.)

20 A chain labeling Define: min B = least element of B j < B : j < k k B. Suppose π i is obtained from π i 1 by merging together blocks B and B, with min B < min B. Define Λ i (m) = max{j B : j < B } Λ(m) = (Λ 1 (m),...,λ n (m)).

21 An example we have Λ(m) = (2, 3, 1, 2).

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

23 Number of chains Theorem. Λ is a bijection between the maximal chains of noncrossing partitions of {1,...,n + 1} and parking functions of length n. Corollary (Kreweras, 1972). The number of maximal chains of noncrossing partitions of {1,...,n + 1} is (n + 1) n 1.

24 Number of chains Theorem. Λ is a bijection between the maximal chains of noncrossing partitions of {1,...,n + 1} and parking functions of length n. Corollary (Kreweras, 1972). The number of maximal chains of noncrossing partitions of {1,...,n + 1} is (n + 1) n 1. Is there a connection with Voiculescu s theory of free probability?

25 The Shi arrangement: background Braid arrangement B n : the set of hyperplanes in R n. x i x j = 0, 1 i < j n,

26 The Shi arrangement: background Braid arrangement B n : the set of hyperplanes in R n. x i x j = 0, 1 i < j n, R = set of regions of B n #R =??

27 The Shi arrangement: background Braid arrangement B n : the set of hyperplanes in R n. x i x j = 0, 1 i < j n, R = set of regions of B n #R = n!

28 The Shi arrangement: background Braid arrangement B n : the set of hyperplanes in R n. x i x j = 0, 1 i < j n, R = set of regions of B n #R = n! Let R 0 be the base region R 0 : x 1 > x 2 > > x n.

29 Labeling the regions Label R 0 with λ(r 0 ) = (1, 1,..., 1) Z n. If R is labelled, R is separated from R only by x i x j = 0 (i < j), and R is unlabelled, then set λ(r ) = λ(r) + e i, where e i = ith unit coordinate vector.

30 The labeling rule R R λ( ) R λ( R )=λ( R)+e i x i = x i < j j

31 Description of labels x =x 1 2 x =x 1 3 B 3 x =x 2 3

32 Description of labels x =x 1 2 x =x 1 3 B 3 x =x 2 3 Theorem (easy). The labels of B n are the sequences (b 1,...,b n ) Z n such that 1 b i n i + 1.

33 Separating hyperplanes Recall R 0 : x 1 > x 2 > > x n. Let d(r) be the number of hyperplanes in B n separating R 0 from R.

34 The case n = x =x x =x x 2 =x3 1 3 B 3

35 Generating function for d(r) NOTE: If λ(r) = (b 1,...,b n ), then d(r) = (b i 1).

36 Generating function for d(r) NOTE: If λ(r) = (b 1,...,b n ), then d(r) = (b i 1). Easy consequence: Corollary. R R q d(r) = (1 + q)(1 + q + q 2 ) (1 + q + + q n 1 ).

37 The Shi arrangement Shi Jianyi ( )

38 The Shi arrangement Shi Jianyi ( ) Shi arrangement S n : the set of hyperplanes x i x j = 0, 1, 1 i < j n, in R n.

39 The case n = 3 x -x =1 x -x = x -x =0 1 2 x -x =1 1 2 x -x =1 x -x =

40 Labeling the regions base region: R 0 : x n + 1 > x 1 > > x n

41 Labeling the regions base region: R 0 : x n + 1 > x 1 > > x n λ(r 0 ) = (1, 1,..., 1) Z n

42 If R is labelled, R is separated from R only by x i x j = 0 (i < j), and R is unlabelled, then set λ(r ) = λ(r) + e i. If R is labelled, R is separated from R only by x i x j = 1 (i < j), and R is unlabelled, then set λ(r ) = λ(r) + e j.

43 The labeling rule R λ( ) R R λ( R )=λ( R)+e i R λ( ) R R λ( )=λ( )+e R R j x i = x i < j j x i = x i < j j +1

44 The labeling for n = x x =1 x x = x 1 x 2 = x x =0 x x =1 x x =0 1 2

45 Description of the labels Theorem (Pak, S.). The labels of S n are the parking functions of length n (each occurring once).

46 Description of the labels Theorem (Pak, S.). The labels of S n are the parking functions of length n (each occurring once). Corollary (Shi, 1986) r(s n ) = (n + 1) n 1

47 The parking function S n -module The symmetric group S n acts on the set P n of all parking functions of length n by permuting coordinates.

48 Sample properties Multiplicity of trivial representation (number of orbits) = C n = n+1( 1 2n ) n n = 3 : Number of elements of P n fixed by w S n (character value at w): (#cycles of w) 1 #Fix(w) = (n + 1)

49 Symmetric functions For symmetric function aficionados: Let PF n = ch(p n ). PF n = λ n (n + 1) l(λ) 1 z 1 λ p λ = λ n = λ n 1 n + 1 s λ(1 n+1 )s λ [ 1 ( ) ] λi + n m λ n + 1 n i

50 More properties PF n ωpf n = λ n = λ n n(n 1) (n l(λ) + 2) h λ. m 1 (λ)! m n (λ)! [ 1 ( ) ] n + 1 m λ. n + 1 i λ i

51 Background: invariants of S n The group S n acts on R = C[x 1,...,x n ] by permuting variables, i.e., w x i = x w(i). Let R S n = {f R : w f = f for all w S n}.

52 Background: invariants of S n The group S n acts on R = C[x 1,...,x n ] by permuting variables, i.e., w x i = x w(i). Let R S n = {f R : w f = f for all w S n}. Well-known: R S n = C[e 1,...,e n ], where e k = 1 i 1 <i 2 < <i k n x i1 x i2 x ik.

53 The coinvariant algebra Let D = R/ ( R S n + ) = R/(e 1,...,e n ). Then dim D = n!, and S n acts on D according to the regular representation.

54 Diagonal action of S n Now let S n act diagonally on i.e, As before, let R = C[x 1,...,x n,y 1,...,y n ], w x i = x w(i), w y i = y w(i). R S n = {f R : w f = f for all w S n} ( ) D = R/. R S n +

55 Hainman s theorem Theorem (Haiman, 1994, 2001). dim D = (n + 1) n 1, and the action of S n on D is isomorphic to the action on P n, tensored with the sign representation. (Connections with Macdonald polynomials, Hilbert scheme of points in the plane, etc.)

56 λ-parking functions Let λ = (λ 1,...,λ n ), λ 1 λ n > 0. A λ-parking function is a sequence (a 1,...,a n ) P n whose increasing rearrangement b 1 b n satisfies b i λ n i+1.

57 λ-parking functions Let λ = (λ 1,...,λ n ), λ 1 λ n > 0. A λ-parking function is a sequence (a 1,...,a n ) P n whose increasing rearrangement b 1 b n satisfies b i λ n i+1. Ordinary parking functions: λ = (n,n 1,...,1)

58 Enumeration of λ-parking fns. Theorem (Steck 1968, Gessel 1996). The number N(λ) of λ-parking functions is given by N(λ) = n! det [ λ j i+1 n i+1 (j i + 1)! ] n i,j=1

59 The parking function polytope Given x 1,...,x n R 0, define P = P(x 1,..., x n ) R n by: (y 1,...,y n ) P n if for 1 i n. 0 y i, y y i x x i

60 The parking function polytope Given x 1,...,x n R 0, define P = P(x 1,..., x n ) R n by: (y 1,...,y n ) P n if 0 y i, y y i x x i for 1 i n. (also called Pitman-Stanley polytope)

61 Volume of P Theorem. (a) Let x 1,...,x n N. Then n!v (P n ) = N(λ), where λ n i+1 = x x i. (b) n!v (P n ) = x i1 x in. parking functions (i 1,...,in)

62 Volume of P Theorem. (a) Let x 1,...,x n N. Then n!v (P n ) = N(λ), where λ n i+1 = x x i. (b) n!v (P n ) = x i1 x in. parking functions (i 1,...,in) NOTE. If each x i > 0, then P n has the combinatorial type of an n-cube.

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