PARKING FUNCTIONS. Richard P. Stanley Department of Mathematics M.I.T Cambridge, MA
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1 PARKING FUNCTIONS Richard P. Stanley Department of Mathematics M.I.T. -75 Cambridge, MA 09 Transparencies available at:
2 ENUMERATION OF PARKING FUNCTIONS... n... a a a n Car C i prefers space a i. If a i is occupied, then C i takes the next available space. We call (a ; : : : ; a n ) a parking function (of length n) if all cars can park. n = : n = :
3 Easy: Let = (a ; : : : ; a n ) P n. Let b b b n be the increasing rearrangement of. Then 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 Theorem (Pyke, 959; Konheim and Weiss, 966). Let f(n) be the number of parking functions of length n. Then f(n) = (n + ) n. Proof (Pollak, c. 974). Add an additional space n +, and arrange the spaces in a circle. Allow n + also as a preferred space. n+ a a a... n n 4
5 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 +. If = (a ; : : : ; a n ) leads to car C i parking at space p i, then (a + j; : : : ; a n + j) (modulo n + ) will lead to car C i parking at space p i +j. Hence exactly one of the vectors (a +i; a +i; : : : a n +i) (modulo n + ) is a parking function, so f(n) = (n + )n n + = (n + ) n : 5
6 FOREST INVERSIONS Let F be a rooted forest on the vertex set f; : : : ; ng THEOREM (Sylvester-Borchardt- Cayley). The number of such forests is (n + ) n. 6
7 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
8 inv(f ) = #(inversions of F ) Inversions: (5; 4); (5; ); (; 4); (; 8) (; ); (0; ); (0; 6); (0; 9) inv(f ) = 8 8
9 Let I n (q) = X F q inv(f ) ; summed over all forests F with vertex set f; : : : ; ng. E.g., I (q) = I (q) = + q I (q) = 6 + 6q + q + q Theorem (Mallows-Riordan 968, Gessel- Wang 979) We have I n ( + q) = X G q e(g) n ; where G ranges over all connected graphs (without loops or multiple edges) on n+ labelled vertices, and where e(g) denotes the number of edges of G. 9
10 Corollary. X n0 P I n (q)(q ) nxn n! = n0 q(n+ )xn n! Pn0 q(n )xn n! X n X I n (q)(q ) n xn n! = log q (n )xn n! n0 Theorem (Kreweras, 980) We have q (n ) I n (=q) = X (a ;:::;an) q a ++an ; where (a ; : : : ; a n ) ranges over all parking functions of length n. 0
11 NONCROSSING PARTITIONS A noncrossing partition of f; ; : : : ; ng is a partition fb ; : : : ; B k g of f; : : : ; ng such that a < b < c < d; a; c B i ; b; d B j ) i = j:
12 Theorem (H. W. Becker, 948{49) The number of noncrossing partitions of f; : : : ; ng is the Catalan number C n = n + n n :
13 A maximal chain m of noncrossing partitions of f; : : : ; n+g is a sequence 0 ; ; ; : : : ; n of noncrossing partitions of f; : : : ; n + g such that i is obtained from i by merging two blocks into one. (Hence i has exactly n + i blocks.)
14 Dene: min B = least element of B j < B : j < k 8k 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 B : j < B 0 g (m) = ( (m); : : : ; n (m)): For above example: we have (m) = (; ; ; ): 4
15 Theorem. is a bijection between the maximal chains of noncrossing partitions of f; : : : ; n + g and parking functions of length n. Corollary (Kreweras, 97) The number of maximal chains of noncrossing partitions of f; : : : ; n + g is (n + ) n : Is there a connection with Voiculescu's theory of free probability? 5
16 THE SHI ARRANGEMENT Braid arrangement B n : the set of hyperplanes in R n. x i x j = 0; i < j n; R = set of regions of B n #R = n! Let R 0 be the \base region" R 0 : x > x > > x n : Let d(r) be the number of hyperplanes in B n separating R 0 from R. 6
17 0 x =x x =x x =x B 7
18 Proposition. #fr : d(r) = jg = #fw S n : `(w) = where n jg; `(w) = #f(r; s) : r < s; w(r) > w(s)g; the number of inversions of w. X RR q d(r) = (+q)(+q+q ) (+q+ +q n ): 8
19 Label R 0 with (R 0 ) = (; ; : : : ; ) Z n : If R is labelled, R 0 is separated from R only by x i x j = 0 (i < j), and R 0 is unlabelled, then set (R 0 ) = (R) + e i ; where e i = ith unit coordinate vector. R λ( ) R R λ( R )=λ( R)+e i x i = x i < j j 9
20 x =x x =x B x =x Theorem (easy). The labels of B n are the sequences (a ; : : : ; a n ) Z n such that a i n i +. 0
21 Shi arrangement S n : the set of hyperplanes x i x j = 0; ; i < j n; in R n : x -x = x -x =0 x -x =0 x -x = x -x = x -x =0
22 base region R 0 : x n + > x > > x n (R 0 ) = (; ; : : : ; ) Z n If R is labelled, R 0 is separated from R only by x i x j = 0 (i < j), and R 0 is unlabelled, then set (R 0 ) = (R) + e i : If R is labelled, R 0 is separated from R only by x i x j = (i < j), and R 0 is unlabelled, then set (R 0 ) = (R) + e j :
23 R λ( ) R R λ( R )=λ( R)+e i R λ( ) R R λ( R )=λ( R)+ej x i = x i < j j x i = x i < j j +
24 x -x = x -x =0 x -x =0 x -x = x -x = x -x =0 Theorem (Pak, S.). The labels of S n are the parking functions of length n (each occurring once). 4
25 Corollary (Shi, 986) r(s n ) = (n + ) n As for B n, let d(r) be the number of hyperplanes in S n separating R 0 and R. Note: If (R) = (a ; : : : ; a n ), then d(r) = a + + a n n: Let R = set of regions of S n. Corollary. X RR q d(r) = q (n ) I n (=q): 5
26 THE PARKING FUNCTION S n -MODULE The symmetric group acts on the set P n of all parking functions of length n by permuting coordinates. Sample properties: Multiplicity of trivial representation (number of orbits) = C n = n+ n n n = : Number of elements of P n xed by w S n (character value at w): (# cycles of w) #Fix(w) = (n + ) 6
27 For symmetric function acionados: Let PF n = ch(p n ). PF n = X `n (n + )`() z p X = `n X = `n X = `n X!PF n = `n n + s ( n+ )s n + 4 Y i i + n n 5 m n(n ) (n `() + ) n + m ()! m n ()! 4 Y i n + i 5 m : h : 7
28 Moreover X PF n t n+ = (te( t)) h i ; n0 where E(t) = P n0 e nt n, and h i denotes compositional inverse. 8
29 THE HAIMAN MODULE The group S n acts on R = C [x ; : : : ; x n ] by permuting variables, i.e., w x i = x w(i). Let R Sn = ff R : wf = f for all w S n g: Well-known: where e k = Let D = R= R Sn = C [e ; : : : ; e n ]; X i <i <<i k n x i x i x i k : R S n + = R=(e ; : : : ; e n ): Then dim D = n!, and S n acts on D according to the regular representation. 9
30 i.e, Now let S n act diagonally on R = C [x ; : : : ; x n ; y ; : : : ; y n ]; w x i = x w(i) ; w y i = y w(i) : As before, let R Sn = ff R : w f = f for all w S n g D = R= R S n + : Conjecture (Haiman, 994). dim D = (n + ) n, 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.) 0
31 A GENERALIZATION Let = ( ; : : : ; n ); n > 0: A -parking function is a sequence (a ; : : : ; a n ) P n whose increasing rearrangement b b n satises b i n i+. Ordinary parking functions: = (n; n ; : : : ; ) Number (Steck 968, Gessel 996): N() = n! det 4 j i+ n i+ (j i + )! n 5 i;j=
32 The Parking Function Polytope (with J. Pitman) Given x ; : : : ; x n R 0, dene P = P(x ; : : : ; x n ) R n by: (y ; : : : ; y n ) P n if 0 y i ; y + + y i x + + x i for i n. Theorem. (a) Let x ; : : : ; x n N. Then n! V (P n ) = N(); where n i+ = x + + x i. (b) n! V (P n ) = X parking functions (i ;:::;in) x i x i n. Note. If each x i > 0, then P n has the combinatorial type of an n-cube.
33 REFERENCES. H. W. Becker, Planar rhyme schemes, Bull. Amer. Math. Soc. 58 (95), 9. Math. Mag. (948{49), {6. P. H. Edelman, Chain enumeration and non-crossing partitions, Discrete Math. (980), 7{80.. P. H. Edelman and R. Simion, Chains in the lattice of noncrossing partitions, Discrete Math. 6 (994), 07{9. 4. D. Foata and J. Riordan, Mappings of acyclic and parking functions, aequationes math. 0 (974), 0{. 5. J. Francon, Acyclic and parking functions, J. Combinatorial Theory (A) 8 (975), 7{5. 6. I. Gessel and D.-L. Wang, Depth-rst search as a combinatorial correspondence, J. Combinatorial Theory (A) 6 (979), 08{. 7. M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combinatorics (994), 7{ P. Headley, Reduced expressions in innite Coxeter groups, Ph.D. thesis, University of Michigan, Ann Arbor, P. Headley, On reduced expressions in ane Weyl groups, in Formal Power Series and Algebraic Combinatorics, FP- SAC '94, May {7, 994, DIMACS preprint, pp. 5{. 0. A. G. Konheim and B. Weiss, An occupancy discipline and applications, SIAM J. Applied Math. 4 (966), 66{74.. G. Kreweras, Sur les partitions non croisees d'un cycle, Discrete Math. (97), {50.. G. Kreweras, Une famille de polyn^omes ayant plusieurs proprietes enumeratives, Periodica Math. Hung. (980), 09{0.
34 . J. Lewis, Parking functions and regions of the Shi arrangement, preprint dated August C. L. Mallows and J. Riordan, The inversion enumerator for labeled trees, Bull Amer. Math. Soc. 74 (968), 9{ Y. Poupard, Etude et denombrement paralleles des partitions non croisees d'un cycle et des decoupage d'un polygone convexe, Discrete Math. (97), 79{ R. Pyke, The supremum and inmum of the Poisson process, Ann. Math. Statist. 0 (959), 568{ V. Reiner, Non-crossing partitions for classical reection groups, Discrete Math. 77 (997), 95{. 8. J.-Y. Shi, The Kazhdan-Lusztig cells in certain ane Weyl groups, Lecture Note in Mathematics, no. 79, Springer, Berlin/Heidelberg/New York, J.-Y. Shi, Sign types corresponding to an ane Weyl group, J. London Math. Soc. 5 (987), 56{ R. Simion, Combinatorial statistics on non-crossing partitions, J. Combinatorial Theory (A) 66 (994), 70{0.. R. Simion and D. Ullman, On the structure of the lattice of noncrossing partitions, Discrete Math. 98 (99), 9{06.. R. Speicher, Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann. 98 (994), 6{64.. R. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Nat. Acad. Sci. 9 (996), 60{ R. Stanley, Parking functions and noncrossing partitions, Electronic J. Combinatorics 4, R0 (997), 4 pp. 5. R. Stanley, Hyperplane arrangements, parking functions, and tree inversions, in Mathematical Essays in Honor of 4
35 Gian-Carlo Rota (B. Sagan and R. Stanley, eds.), Birkhauser, Boston/Basel/Berlin, 998, pp. 59{ G. P. Steck, The Smirnov two-sample tests as rank tests, Ann. Math. Statist. 40 (968), C. H. Yan, Generalized tree inversions and k-parking functions, J. Combinatorial Theory (A) 79 (997), 68{80. 5
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