Affine permutations and rational slope parking functions Eugene Gorsky, Mikhail Mazin, Monica Vazirani To cite this version: Eugene Gorsky, Mikhail Mazin, Monica Vazirani. Affine permutations and rational slope parking func- tions. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.887-898. hal-01207556 HAL Id: hal-01207556 https://hal.inria.fr/hal-01207556 Submitted on 1 Oct 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. FPSAC 2014, Chicago, USA DMTCS proc. AT, 2014, 887–898 Affine permutations and rational slope parking functions Eugene Gorsky1 Mikhail Mazin2 Monica Vazirani3 1 Columbia University, New York, New York, USA 2 Kansas State University, Manhattan, Kansas, USA 3 University of California at Davis, Davis, California, USA Abstract. We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our construction to two previously known combinatorial constructions: Haglund’s bijection ζ exchanging the pairs of statistics (area; dinv) and (bounce; area) on Dyck paths, and Pak-Stanley labeling of the regions of k-Shi hyperplane arrangements by k-parking functions. Essentially, our approach can be viewed as a generalization and a unification of these two constructions. Resum´ e.´ Nous introduisons une nouvelle approche pour enum´ erer´ les fonctions parking pente rationelles, en se focalisant sur la statistique area ainsi qu’une gen´ eralisation´ de la statistique dinv. Nous rapprochons la combinatoire des fonctions de parking a` celle des permutations affines. Nous ramenons la construction a` deux constructions dej´ a` connues: la bijection d’Haglund ζ entre les statistiques (area; dinv) et (bounce; area) sur les chemins de Dyck, et l’etiquetage´ de Pak-Stanley des arrangements des k-Shi hyperplans par les k-fonctions de parking. Essentiellement, notre approche gen´ eralise´ et unifie ces deux constructions. Keywords: Parking functions, affine permutations 1 Introduction Parking functions are prevalent in modern combinatorics. There is a natural action of the symmetric group on parking functions, and the orbits are labeled by the non-decreasing parking functions which correspond naturally to Dyck paths. This provides a link between parking functions and various combinatorial objects counted by Catalan numbers. In a series of papers Garsia, Haglund, Haiman, et al. [10, 11], related the combinatorics of Catalan numbers and parking functions to the space of diagonal harmonics. There are also deep connections to the geometry of the Hilbert scheme. Since the works of Pak and Stanley [16], Athanasiadis and Linusson [4] , it became clear that park- ing functions are tightly related to the combinatorics of the affine symmetric group. In particular, they provided two different bijections between the parking functions and the regions of Shi hyperplane ar- rangement. It has been remarked in [2, 6, 15] that the inverses of the affine permutations labeling the n+1 minimal alcoves in Shi regions belong to a certain simplex Dn , which is isometric to the (n+1)-dilated 1365–8050 © 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France 888 Eugene Gorsky, Mikhail Mazin, Monica Vazirani 2 f ∈ PF7~4 4 1 f ∉ PF5~4 3 Fig. 1: The labeled diagram for the parking function f = 2040 : L M n+1 fundamental alcove. As a result, the alcoves in Dn can be labeled by parking functions in two different ways. Many of the concepts above have “rational” counterparts corresponding to a coprime pair (m; n), for which the classical case is (n+1; n). In [8] and [9] the first and the second authors proved that the rational Dyck paths label the affine cells in a certain algebraic variety, the compactified Jacobian of a plane curve singularity with one Puiseaux pair (m; n). This work was generalized by Hikita in [12] who proved that m~n parking functions naturally label the cells in a certain Springer fibre in the affine flag variety for the affine symmetric group S̃n. As a consequence of this paper, we can use our construction to prove new formulas for the Poincare´ polynomials of certain affine Springer fibers relying on the work of [8] and [9]. The present paper is dedicated to the systematic study of the m~n parking functions in terms of the m affine symmetric group S̃n. We consider the set S̃n of all affine permutations in S̃n with no inversions m of height m and two maps A and PS from S̃n to the set of m~n parking functions PF m~n. When one −1 composes PS with A , one recovers the zeta map ζ of Haglund, which was used to give a combinatorial proof of the weak symmetry property of the q; t-Catalan numbers. Acknowledgements The authors would like to thank the American Institute of Mathematics for their hospitality, and D. Arm- strong, F. Bergeron, S. Fishel, I. Pak, R. Stanley, V. Reiner, B. Rhoades, A. Varchenko, G. Warrington and N. Williams for useful discussions and suggestions. E. G. was partially supported by the grants RFBR-13- 01-00755, NSh-4850.2012.1. M. V. was partially supported by the grant NSA MSP H98230-12-1-0232. 2 Tools and definitions We start with a brief review of the definitions and basic results involving parking functions, affine permu- tations, and hyperplane arrangements, which will play the key role in our constructions. 2.1 Parking Functions Definition 2.1 A function f ∶ {1; : : : ; n} → Z≥0 is called an m~n-parking function if the Young diagram with row lengths equal to f(1); : : : ; f(n) put in decreasing order, bottom to top, fits under the diagonal in an n × m rectangle. The set of such functions is denoted by PF m~n : We will often use the notation f = f(1)f(2) : : : f(n) for parking functions. L M Example 2.2 Consider the function f ∶ {1; 2; 3; 4} → Z≥0 given by f(1) = 2; f(2) = 0; f(3) = 4; and f(4) = 0 (i.e. f = 2040 ). The corresponding Young diagram fits under the diagonal in a 4 × 7 rectangle, L M but does not fit under the diagonal in a 4 × 5 rectangle. Thus, f ∈ PF7~4 but f ∉ PF5~4 (see Figure 1). Affine permutations and rational slope parking functions 889 Equivalently, a function f ∶ {1; : : : ; n} → Z≥0 belongs to PF m~n if and only if it satisfies im i 0; : : : ; n 1 ; k 1; : : : ; n f k i 1: ∀ ∈ { − } ♯{ ∈ { }S ( ) ≤ n } ≥ + Let P ∶ PF m~n → Ym;n denote the natural map from the set of parking functions to the set Ym;n of Young diagrams that fit under diagonal in an n × m rectangle. To recover a parking function f ∈ PF m~n from the corresponding Young diagram P (f) one needs some extra information. Lengths of the rows of P (f) correspond to the values of f; but one needs also to assign the preimages to them. That is, one should label the rows of P (f) by integers 1; 2; : : : ; n: Note that if P (f) has two rows of the same length, then the order of the corresponding labels does not matter. One should choose one of the possible orders. We choose the decreasing order (read from bottom to top). Definition 2.3 Let Ŷm;n denote the set of couples (D; τ) of a Young diagram D ∈ Ym;n and a (finite) permutation τ ∈ Sn; such that if kth and (k + 1)th rows of D have the same length, then τ(k + 1) < τ(k): We will refer to τ as the row-labeling of D: Note that τ ∈ Sn is the permutation of maximal length such that f ○ τ is non-increasing. Example 2.4 In Example 2.2, one has τ = [3; 1; 4; 2], so f ○ τ = 2040 ○ [3; 1; 4; 2] = 4200 . L M L M We get the following lemma: Lemma 2.5 The set of m~n-parking functions PF m~n is in bijection with the set of labeled Young dia- grams Ŷm;n: Remark 2.6 Note that for m = n + 1 the set PF m~n is exactly the set of classical parking functions PF, and for m = kn + 1 it is the set of k-parking functions PF k (e.g. [10]). From now on we will assume that m and n are coprime, so there are no lattice points on the diagonal of n × m rectangle. By abuse of notation, we will call a non-decreasing parking function increasing. The 1 n+m number of increasing parking functions equals the generalized Catalan number ♯ Ym;n = n+m n . The n−1 number of all parking functions ♯ PF m~n equals m . (See [4].) 2.2 Affine Permutations Definition 2.7 The affine symmetric group S̃n is generated by s1; : : : ; sn−1; s0 subject to the relations 2 si = 1; sisj = sjsi for i − j ≡~ ±1 mod n; sisjsi = sjsisj for i − j ≡ ±1 mod n (if n > 2): Let x1 ⎛ ⎞ n n x = ⎜ ⋮ ⎟ ;V ∶= {x ∈ R S x1 + ::: + xn = 0} ⊂ R ⎝ xn ⎠ k k and let Hij be the hyperplane {x ∈ V S xi − xj = k} ⊂ V: The hyperplane arrangement B̃n = {Hij ∶ 0 < i < j ≤ n; k ∈ Z} is called the affine braid arrangement.