aa r X i v : . [ m a t h . C O ] N ov Discrete Mathematics and Theoretical Computer Science
DMTCS vol. (subm.) , by the authors, 1–1
Type C parking functions and a zeta map Robin Sulzgruber † and Marko Thiel ‡ Fakult¨at f¨ur Mathematik, Universit¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria received 2014-11-14 , revised tba , accepted tba . Abstract
We introduce type C parking functions, encoded as vertically labelled lattice paths and endowed with astatistic dinv’ . We define a bijection from type C parking functions to regions of the Shi arrangement of type C ,encoded as diagonally labelled ballot paths and endowed with a natural statistic area’ . This bijection is a naturalanalogue of the zeta map of Haglund and Loehr and maps dinv’ to area’ . We give three different descriptions of it. R´esum´e
Nous introduisons les fonctions de stationnement de type C , encod´ees par des chemins ´etiquet´es verticale-ment et munies d’une statistique dinv’ . Nous d´efinissons une bijection entre les fonctions de stationnement de type C et les r´egions de l’arrangement de Shi de type C , encod´ees par des chemins ´etiquet´es diagonalement et munies d’unestatistique naturelle area’ . Cette bijection est un analogue naturel `a la fonction zeta de Haglund et Loehr, et envoie dinv’ sur area’ . Nous donnons trois diff´erentes descriptions de celle-ci. Keywords: parking functions, Shi arrangement, zeta map, dinv statistic
One of the most well-studied objects in algebraic combinatorics is the space of diagonal harmonics of thesymmetric group S n . Its Hilbert series has two (conjectural) combinatorial interpretations: DH ( n ; q, t ) = X P ∈ Park n q dinv’( P ) t area( P ) = X R ∈ Diag n q area’( R ) t bounce( R ) , where Park n is the set of parking functions of length n , viewed as vertically labelled Dyck paths, and Diag n is the set of diagonally labelled Dyck paths with n steps. There is a bijection ζ due to Haglundand Loehr (2005) that maps Park n to Diag n and sends the bistatistic (dinv’ , area) to (area’ , bounce) ,demonstrating the second equality.The combinatorial objects Park n and Diag n may be viewed as the type A n − cases of more gen-eral objects associated to any crystallographic root system Φ . These are, respectively, the finite torus † Email: [email protected] . ‡ Email: [email protected] .Research supported by the Austrian Science Fund (FWF), grant S50-N15 in the framework of the Special Research Program “Algo-rithmic and Enumerative Combinatorics” (SFB F50).subm. to DMTCS c (cid:13) by the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
Robin Sulzgruber and Marko Thiel ˇ Q/ ( h + 1) ˇ Q and the set of regions of the Shi arrangement of Φ . Here ˇ Q is the coroot lattice and h is theCoxeter number of Φ . Both of these objects have the same cardinality ( h + 1) r , where r is the rank of Φ , so there should be a uniform “zeta map” giving a bijection between them. This map does in fact exist,and will be described in future work.In the present extended abstract we focus on the root system of type C n . In Section 2 we present thenecessary background on Weyl groups and the Shi arrangement. In Section 3 we introduce combinatorialmodels for the finite torus of type C in terms of vertically labelled lattice paths and for the set of regionsof the Shi arrangement of type C in terms of diagonally labelled ballot paths . We also introduce a statistic dinv’ on vertically labelled lattice paths and a statistic area’ on diagonally labelled ballot paths. Thesestatistics are natural analogues of the corresponding statistics in type A .In Section 4 we describe a map between these two combinatorial models that we call the type C zetamap . We give three descriptions of this map, all similar in style to different descriptions of the classicalzeta map. The first description in terms of area vectors follows Haglund and Loehr (2005). The seconddescription in terms of ascents and valleys resembles that of (Armstrong et al., 2014a, Section 5.2). Thethird description as a sweep map is in the spirit of Armstrong et al. (2014b). Our main result (Theorem 4.2)is that the zeta map of type C is a bijection that sends the dinv’ statistic to the area’ statistic.We prioritise examples and prefer to include an adequate presentation of the known combinatorialobjects of type A rather than presenting proofs. A full version of this extended abstract containing allproofs is in preparation. Let Φ be an irreducible crystallographic root system of rank r , with simple system ∆ = { α , α , . . . , α r } ,positive system Φ + and ambient space V . For background on root systems and reflection groups seeHumphreys (1990). For α ∈ Φ , let s α be the reflection in the hyperplane H α = { x ∈ V : h x, α i = 0 } . Then the Weyl group W of Φ is the group of automorphisms of V generated by all the s α with α ∈ Φ .Define the Coxeter arrangement of Φ as the central hyperplane arrangement in V given by all the hyper-planes H α for α ∈ Φ . The connected components of the complement of the union of these hyperplanesare called chambers . The Weyl group W acts simply transitively on the chambers, so if we define the dominant chamber as C = { x ∈ V : h x, α i > for all α ∈ ∆ } , we may write every chamber as wC for a unique w ∈ W .For α ∈ Φ and d ∈ Z , let s dα be the reflection in the affine hyperplane H dα = { x ∈ V : h x, α i = d } . Then the affine Weyl group f W of Φ is the group of affine transformations of V generated by all the s dα for α ∈ Φ and d ∈ Z . Define the affine Coxeter arrangement as the affine hyperplane arrangement in V ype C parking functions and a zeta map H dα for α ∈ Φ and d ∈ Z . The connected components of the complement of the union ofthese hyperplanes are called alcoves . The affine Weyl group f W acts simply transitively on the alcoves, soif we write ˜ α for the highest root of Φ and define the fundamental alcove as A ◦ = { x ∈ V : h x, α i > for all α ∈ ∆ and h x, ˜ α i < } , we may write every alcove as w a A ◦ for a unique w a ∈ f W . The affine Weyl group f W acts on the corootlattice ˇ Q , and if we identify ˇ Q with its translation group we may write f W = W ⋉ ˇ Q as a semidirectproduct.If α ∈ Φ + and w a ∈ f W , there is a unique integer k such that k < h x, α i < k + 1 for all x ∈ w a A ◦ .We denote this integer by k ( w a , α ) . Define the
Shi arrangement as the hyperplane arrangement given by the hyperplanes H dα for α ∈ Φ + and d = 0 , . Then the complement of the union of these hyperplanes falls apart into connected components,which are called the regions of the arrangement. The hyperplanes that support facets of a region R arecalled the walls of R . Those walls of R that do not contain the origin and separate R from the origin arecalled the floors of R . Define the walls and floors of an alcove similarly. Notice that every wall of a regionis a hyperplane of the Shi arrangement, but the walls of an alcove need not be. We call a region or alcove dominant if it is contained in the dominant chamber. Theorem 2.1 (Shi, 1987, Prop 7.1)
Every region R of the Shi arrangement has a unique minimal alcove w R A ◦ ⊆ R , which is the alcove in R closest to the origin. That is, for any α ∈ Φ + and w a ∈ f W suchthat w a A ◦ ⊆ R , we have | k ( w R , α ) | ≤ | k ( w a , α ) | . We define W Shi = { w R : R is a Shi region } . The corresponding alcoves w R A ◦ we call Shi alcoves .That is, we call an alcove a Shi alcove if it is the minimal alcove of the Shi region containing it.
Theorem 2.2 (Shi, 1987, Prop 7.3)
The alcove w a A ◦ is a Shi alcove if and only if all floors of w a A ◦ arehyperplanes of the Shi arrangement. The following theorem is already known for dominant regions (Athanasiadis, 2005, Prop 3.11).
Theorem 2.3
The floors of the minimal alcove w R A ◦ of a Shi region R are exactly the floors of R . The following lemma describes what the Shi arrangement looks like in each chamber.
Lemma 2.4 (Armstrong et al., 2012, Lemma 10.2)
For w ∈ W , the hyperplanes of the Shi arrangementthat intersect the chamber wC are exactly those of the form H w ( α ) where α ∈ Φ + and w ( α ) ∈ Φ + . Thus if w R A ◦ is a Shi alcove contained in the Weyl chamber wC , then by Theorem 2.2 and Lemma 2.4all its floors are of the form H w ( α ) where α ∈ Φ + and w ( α ) ∈ Φ + . So w − w R A ◦ is a dominant alcoveand its floors are of the form w − ( H w ( α ) ) = H α with α ∈ Φ + . It is thus a Shi alcove by Theorem 2.2.Conversely, if w R A ◦ is a dominant Shi alcove and w ∈ W , then ww R A ◦ is a Shi alcove if and only if w ( α ) ∈ Φ + whenever H α is a floor of w R A ◦ . Thus the map Θ : w R ( w − w R , w ) Robin Sulzgruber and Marko Thiel • ◦ • •
Fig. 1:
A ballot path β ∈ B (left) with one valley (1 , and two rises , , and a Dyck path π ∈ D (right) withvalleys (2 , , (3 , and rises , , . We have A β = { e − e , e } and A π = { e − e , e − e } where w R A ◦ ⊆ wC , is a bijection from W Shi to the set of pairs ( w R , w ) such that w R A ◦ is a dominantShi alcove, w ∈ W and w ( α ) ∈ Φ + whenever H α is a floor of w R A ◦ .Define a partial order on Φ + by α ≤ β if and only if β − α can be written as a linear combination ofsimple roots with nonnegative integer coefficients. The set of positive roots Φ + with this partial order iscalled the root poset . It turns out that the map F L : R
7→ { α ∈ Φ + : H α is a floor of R } is a bijection from the set of dominant Shi regions of Φ to the set of antichains in the root poset of Φ . SeeShi (1997). Putting R w R , Θ and F L together and using Theorem 2.3 we get that the map R ( A, w ) , where R ⊆ wC and A = w − ( F L ( R )) , is a bijection from the set of Shi regions to the set of pairs ( A, w ) such that A is an antichain in the root poset, w ∈ W and w ( A ) ⊆ Φ + . A similar bijection using ceilingsinstead of floors is given in (Armstrong et al., 2012, Prop 10.3). If Φ is of type A n − , we take V = { ( x , x , . . . , x n ) ∈ R n : P ni =1 x i = 0 } , Φ = { e i − e j : i = j } and Φ + = { e i − e j : i < j } . The Weyl group W is the symmetric group S n that acts on V by permutingcoordinates, ˇ Q = { ( x , x , . . . , x n ) ∈ Z n : P ni =1 x i = 0 } , r = n − and h = n .If Φ is of type C n , we choose V = R n , Φ = { e i ± e j : i = j } ∪ {± e i : i ∈ [ n ] } and Φ + = { e i ± e j : i > j } ∪ { e i : i ∈ [ n ] } . The Weyl group W is the hyperoctahedral group H n that acts on V by permutingcoordinates and changing signs, ˇ Q = Z n , r = n and h = 2 n . Denote by L m,n the set of lattice paths from (0 , to ( m, n ) consisting of n North steps N = (0 , and m East steps E = (1 , . Let D n denote the set of Dyck paths , that is the subset of L n,n consisting of thepaths that never go below the main diagonal x = y . Let B n denote the set of ballot paths , that is the set oflattice paths starting at (0 , , consisting of n North and/or East steps, and never going below the maindiagonal.A pattern of the form
N N is called rise . A pattern EN is called valley . More precisely, let π be anylattice path with steps s i ∈ { N, E } . We say i is a rise of π if the i -th North step is followed by a Northstep. We say ( i, j ) is a valley of π if the i -th East step is followed by the j -th North step. See Figure 1. ype C parking functions and a zeta map π ∈ D n and Φ is of type A n − , define A π ⊆ Φ + by e i − e j ∈ A π if and only if ( i, j ) is a valley of π . Then the map π A π is a bijection from D n to the set of antichains in the root poset of Φ .If β ∈ B n and Φ is of type C n , define A β ⊆ Φ + by A β = { e i − e j : i > j and ( n + 1 − i, n + 1 − j ) is a valley of β }∪ { e i + e j : i > j and ( n + 1 − i, j + n ) is a valley of β }∪ { e i : the last step of β is its ( n + 1 − i ) -th east step } . Then the map β A β is a bijection from B n to the set of antichains in the root poset of Φ . A diagonally S n -labelled Dyck path is a pair ( π, σ ) of a Dyck path π ∈ D n and a permutation σ ∈ S n such that for each valley ( i, j ) of π we have σ i < σ j . See Figure 2. From the considerations at the end ofSection 2.2, recall that regions of the Shi arrangement of type A n − may be indexed by pairs ( A, σ ) with A an antichain in the root poset, σ ∈ W = S n and σ ( A ) ⊆ Φ + . Proposition 3.1
The map ( π, σ ) ( A π , σ ) is a bijection between diagonally labelled Dyck paths oflength n and regions of the Shi arrangement of type A n − . We provide an interpretation of type C Shi regions as diagonally labelled ballot paths. For any signedpermutation σ ∈ H n we define w σ to be the word of length n given by w σi = σ ( n + 1 − i ) if ≤ i ≤ n and w σi = σ ( n − i ) if n + 1 ≤ i ≤ n . For example if n = 3 then w id = 321¯1¯2¯3 .A diagonally H n -labelled ballot path is a pair ( β, w σ ) of a ballot path β ∈ B n and a word w σ corre-sponding to a signed permutation σ such that for each valley ( i, j ) of β we have w σi > w σj , and such that < w σi if the final step of β is its i -th East step. Hence, if we place the labels w σ in the diagonal then foreach valley the label to its right will be smaller than the label below it. Moreover, if the path ends with anEast step then the label below will be positive. See Figure 3. Proposition 3.2
The map ( β, w σ ) ( A β , σ ) is a bijection between diagonally H n -labelled ballot pathsthe regions of the Shi arrangement of type C n . The area of a Dyck path is defined as the number of boxes strictly between the path and the main diagonal.For example the Dyck path in Figure 2 has area( π ) = 5 . Haglund and Loehr (2005) defined a relatedstatistic area’ for diagonally labelled Dyck paths as follows. Consider a box strictly between the diagonaland the path π in column i and row j . This box contributes to area’( π, σ ) if and only if the label to itsright is larger than the label below it, that is if and only if σ i < σ j . For example the labelled Dyck pathin Figure 2 has area’( π, σ ) = 4 because the nonshaded box in the fifth row and fourth column does notcontribute: σ = 5 > σ = 4 .The area of a ballot path is defined as the number of boxes “below” the path (see Figure 4). We nowdefine a type C area’ statistic on diagonally labelled ballot paths. Let ( β, w σ ) be such a path and considera box below β in column i and row j . This box contributes to area’( β, w σ ) if and only if the label to Robin Sulzgruber and Marko Thiel • •
Fig. 2:
A diagonally labelled Dyck path ( π, σ ) ,where σ = 1 < σ = 2 and σ = 2 < σ = 4 . Wehave area( π ) = 5 and area’( π, σ ) = 4 . − − − ◦• Fig. 3:
A diagonally labelled ballot path ( β, w σ ) ,where w = σ − = − > w = σ − = − and < w = σ − = 2 . We have area( β ) = 4 and area’( β, w σ ) = 1 . Fig. 4:
All ballot paths of length two and their area squares shaded gray. its right is smaller than the label below it, that is if and only if w σi > w σj . The labelled ballot path inFigure 3 has area’( β, w σ ) = 1 since the shaded box in the third row and second column is the only onecontributing: w σ = 2 > w σ = − . For example the box in the fourth row and second column does notcontribute because w σ = 2 < w σ = σ − = 3 .Note that these statistics are the type A and C cases of the following uniform statistic. Define the coheight statistic on regions of the Shi arrangement of any irreducible root system Φ by coheight( R ) = | Φ + | − hyperplanes of the Shi arrangement separating R from the origin . Then the area’ statistics correspond to the coheight statistic under the bijections in Section 3.1.
A vector f = ( f , . . . , f n ) with nonnegative integer entries is called a (classical) parking function oflength n if there exists a permutation σ ∈ S n such that f σ ( i ) ≤ i − for ≤ i ≤ n . Equivalently, f is aparking function if { j : f j ≤ i − } ≥ i for all ≤ i ≤ n .There is a natural S n -isomorphism between the set of parking functions of length n and the finite torus ˇ Q/ ( h + 1) ˇ Q of the root system of type A n − . Thus classical parking functions may be seen as objects oftype A .We define a type C parking function of length n to be an integer vector f = ( f , . . . , f n ) where − n ≤ f i ≤ n for all ≤ i ≤ n . Thus type C parking functions of length n are a natural set ofrepresentatives for the finite torus ˇ Q/ ( h + 1) ˇ Q = Z n / (2 n + 1) Z n of the root system of type C n . ype C parking functions and a zeta map ab a < b a b a b < a ab a < b a b < a ab < a < b Fig. 5:
All six paths in L , and the conditions on their labellings. − − − −
41 2 − − Fig. 6:
Constructing an H -labelled path from the parking function f = (4 , , − , − . Type A parking functions are commonly represented as Dyck paths with labelled North steps (Haglund,2008, Chap. 5). An S n -labelled Dyck path is a pair ( π, σ ) of a Dyck path π ∈ D n and a permutation σ ∈ S n such that σ i < σ i +1 whenever i is a rise of π . Thus, if the label σ i is placed in the box to the rightof the i -th North step then labels increase along columns from bottom to top. For example in Figure 7 wehave σ = 1 < σ = 2 < σ = 4 .We show how type C parking functions can be regarded as labelled lattice paths in a similar fashion.An H n -labelled path ( π, σ ) is a pair of a lattice path π ∈ L n,n and a signed permutation σ ∈ H n suchthat σ i < σ i +1 whenever i is a rise of π and such that < σ if π begins with a North step. Thus, if weplace the label σ i to the left of the i -th North step then the labels increase along columns from bottom totop, and all labels in the zeroth column (that is left of the starting point) are positive. See Figure 5.Given a parking function f = ( f , . . . , f n ) we obtain a labelled path as follows. For all ≤ i ≤ n if f i is non-negative, place the label i in the f i -th column. If f i is negative, place the label − i in column − f i .Rearrange the labels in each column in increasing order and draw a path as in Figure 6.Conversely, let ( π, σ ) be a labelled path. We define a parking function g as follows. If a positive label i occurs in the j -th column then set g i = j . If a negative label i occurs in the j -th column instead set g − i = − j . In summary we have the following result. Proposition 3.3
The above correspondence defines a bijection between type C parking functions of length n and vertically H n -labelled lattice paths. The dinv statistic was first defined by Haiman to provide an (at the time conjectural) combinatorial modelfor the q, t -Catalan numbers (Haglund, 2008, Chap. 3). For each Dyck path π ∈ D n define the areavector ( a , a , . . . , a n ) by letting a i be the number of boxes in the i -th row, strictly between π and themain diagonal. For example the Dyck path in Figure 7 has area vector (0 , , , , . The dinv statistic is Robin Sulzgruber and Marko Thiel
124 3 5
Fig. 7:
A vertically labelled Dyck path ( π, σ ) witharea vector (0 , , , , and dinv( π ) = 5 and dinv’( π, σ ) = 4 . − − Fig. 8:
A vertically labelled path ( β, σ ) with areavector (1 , − , − , , , and dinv( β ) = 9 and dinv’( π, σ ) = 6 . Fig. 9:
Paths with type C area vectors (0 , , , , (1 , , − , and ( − , − , , . defined as dinv( π ) = (cid:8) ( i, j ) : i < j, a i = a j (cid:9) + (cid:8) ( i, j ) : i < j, a i = a j + 1 (cid:9) . A pair ( i, j ) contributing to dinv is called a diagonal inversion . Haglund and Loehr (2005) defined ageneralised statistic dinv’ on vertically S n -labelled Dyck paths. A pair ( i, j ) with a i = a j contributes ifand only if σ i < σ j . On the other hand, a pair ( i, j ) with a i = a j + 1 contributes if and only if σ i > σ j .Compare with Figure 7.We define an area vector and a dinv statistic of type C for lattice paths π ∈ L n,n . The area vector ( a , a , . . . , a n ) is given by a i = i − b i where b i is the number of boxes in the i -th row left of π . SeeFigures 8 and 9. Moreover, we define dinv( π ) = (cid:8) ( i, j ) : i < j, a i = a j (cid:9) + (cid:8) ( i, j ) : i < j, a i = a j + 1 (cid:9) + (cid:8) ( i, j ) : i < j, a i = − a j (cid:9) + (cid:8) ( i, j ) : i < j, a i = − a j + 1 (cid:9) + (cid:8) i : a i = 0 (cid:9) . Next, we define a dinv’ statistic for vertically H n -labelled lattice path ( π, σ ) . As in type A above, a pair ( i, j ) of candidate rows contributes if and only if the labels σ i and σ j satisfy a certain inequality. Moreprecisely, dinv’( π, σ ) = (cid:8) ( i, j ) : i < j, a i = a j , σ i < σ j (cid:9) + (cid:8) ( i, j ) : i < j, a i = a j + 1 , σ i > σ j (cid:9) + (cid:8) ( i, j ) : i < j, a i = − a j , σ i < − σ j (cid:9) + (cid:8) ( i, j ) : i < j, a i = − a j + 1 , σ i > − σ j (cid:9) + (cid:8) i : a i = 0 , σ i < (cid:9) . ype C parking functions and a zeta map a = (0 , , , ,
124 3 5 i = 0 0 111 i = 1 1 1 12 i = 2 2 i = 3 • • Fig. 10:
The classical zeta map: A vertically labelled Dyck path ( π, σ ) (left), the construction of ζ ( π ) (middle), and ζ ( π, σ ) (right). Note that the two definitions of dinv( π ) agree if π is a Dyck path.Consider the H -labelled path ( π, σ ) in Figure 8. Its area vector is given by (1 , − , − , , , . Thereis one diagonal inversion of type a i = a j , namely (1 , , one diagonal inversion of type a i = a j + 1 ,namely (1 , , three diagonal inversions of type a i = − a j , namely (1 , , (2 , and (3 , , three diagonalinversions of type a i = − a j + 1 , namely (1 , , (3 , and (4 , , and one row of length zero, namely i = 4 . In total we have 9 diagonal inversions, so dinv( π ) = 9 . Note that the inversion (1 , is countedtwice!If we wish to take labels into account we find that σ = 2 > , so the row of length zero does notcontribute. Moreover σ = 1 < σ = 2 , thus (1 , is not a d’-inversion of type a i = a j + 1 , and σ = − > − σ = − , thus (2 , is not a d’-inversion of type a i = − a j . The labels of all otherdiagonal inversions fit our requirements, so dinv’( π, σ ) = 6 . The original zeta map is a bijection ζ : D n → D n on Dyck paths and appears in a paper of Andrews et al.(2002). A more explicit treatment including the compatibility with the statistics on Dyck paths defined inthe previous sections can be found in (Haglund, 2008, Thrm. 3.15). Let us start by recalling the definitionof the zeta map.Given a Dyck path π ∈ D n with area vector ( a , a , . . . , a n ) , set i = 0 and place your pen at (0 , .Now read the area vector from left to right drawing an East step for each i − you encounter and a Northstep for each i . Replace i by i + 1 and repeat until you reach the point ( n, n ) . See Figure 10.We describe a bijection ζ C : L n,n → B n which is an analogue of the classical zeta map.Given a path π ∈ L n,n with type C area vector ( a , a , . . . , a n ) , set i = n and start with your pen at (0 , . Read the area vector from left to right drawing an East step for each − i − you encounter and aNorth step for each − i . Then read the area vector from right to left drawing an East step for each i + 1 you encounter and a North step for each i . Now replace i by i − and repeat the process until n stepsare drawn. See Figure 11.It is clear from the construction that ζ C ( π ) never goes below the main diagonal. Moreover when π is aDyck path, then ζ C ( π ) is just the reverse path of ζ ( π ) . In particular ζ C sends Dyck paths to Dyck paths.The following is our first main result.0 Robin Sulzgruber and Marko Thiel − − a = (1 , − , − , , , − i = 2 − − i = 1 − i = 0 5 6 4 3 1 − • • • • ◦ Fig. 11:
The type C zeta map: A vertically labelled lattice path ( π, σ ) (left), the construction of ζ C ( π ) (middle), and ζ ( π, σ ) (right). Note that dinv’( π, σ ) = 6 = area’( ζ ( π, σ )) . Theorem 4.1
The map ζ C : L n,n → B n is a bijection such that dinv C ( π ) = area( ζ C ( π )) . The zeta map can be inverted using the bounce path of a ballot path. A detailed proof will appear in thefull version.
Haglund and Loehr (2005) extended the classical zeta map to a bijection from vertically labelled Dyckpaths to diagonally labelled Dyck paths that sends the dinv’ statistic to the area’ statistic. We start outby recalling their definition. If ( π, σ ) is a vertically S n -labelled path, then ζ ( π, σ ) is simply the diagonallabelling of ζ ( π ) obtained as follows. For i = 0 , , . . . , n read the labels of rows with area equal to i frombottom to top and insert them in the diagonal. Compare with Figure 10.Similarly, in type C we start with a vertically H n -labelled path ( π, σ ) and construct a diagonally labelledballot path ζ ( π, σ ) = ( β, w ) . The ballot path is given by β = ζ C ( π ) . The labelling is obtained as follows.For i = n, n − , . . . , read the labels of the rows with area i from top to bottom and insert them inthe diagonal, then read the labels of rows with area equal to − i + 1 from bottom to top and insert theirnegatives in the diagonal. In the end complement the n labels by adding their negatives in reverse order.See Figures 11, 12 and 13.The theorem below is the main result of this paper. Theorem 4.2
The type C zeta map is a bijection from vertically H n -labelled paths to diagonally H n -labelled ballot paths that sends the dinv’ statistic to the area’ statistic. Combining Theorem 4.2 with Propositions 3.2 and 3.3 we obtain a new proof of the well known resultthat the Shi arrangement of type C n has (2 n + 1) n regions. The Haglund–Loehr zeta map has another simple description given in Armstrong et al. (2014a). Let us fixthe following convention. If ( π, σ ) is a vertically S n -labelled Dyck path with rise i then we say the rise ype C parking functions and a zeta map − ζ ( π, σ ) − − − ◦• ( β, w ) Fig. 12:
The area vector of π is (1 , , . The diagonalinversions are (1 , , (1 , , (1 , , (2 , , i = 2 , and dinv’( π, σ ) = 5 = area’( β, w ) . ( π, σ ) ζ − − − ◦• ( β, w ) Fig. 13:
The area vector of π is (1 , − , . The diag-onal inversions are (1 , , (1 , , (1 , , i = 3 but only (1 , (as inversion of type a i = − a j + 1 ) contributesto dinv’( π, σ ) = 1 . has label ( σ i , σ i +1 ) . If ( π, σ ) is a diagonally labelled Dyck path with valley ( i, j ) then we say the valleyhas label ( σ i , σ j ) .The image ζ ( π, σ ) of a vertically labelled Dyck path under the zeta map can now be defined as follows.First insert the diagonal labelling as described in the previous section. The Dyck path is the unique pathwhich has a valley labelled ( a, b ) if and only if ( π, σ ) has a rise labelled ( a, b ) . See Figure 10.There is a similar description of the zeta map in type C provided by the following proposition. If ( π, σ ) is a vertically H n -labelled path with rise i then we say the rise has label ( σ i , σ i +1 ) . If ( β, w ) is adiagonally labelled ballot path with valley ( i, j ) then we say the valley has label ( w i , w j ) . Proposition 4.3
Let ( π, σ ) be a vertically H n -labelled lattice path. Then ζ ( π, σ ) has valley labelled ( a, b ) if and only if ( π, σ ) has a rise labelled ( b, a ) or ( − a, − b ) . Moreover, ζ ( π, σ ) ends with an East step inthe same column as label a if and only if ( π, σ ) begins with a North step labelled a . Compare with Figures 11, 12 and 13.
13 1 − −
23 30 180 − − − − − − − − − − − − − − − − − − − − − − − − − − − Fig. 14:
The labelling of the steps of a path π (left), the set X of labelled steps (middle), and the path sw( π ) of stepsin increasing order (right). Robin Sulzgruber and Marko Thiel
A generalisation of the zeta map to rational Dyck paths called the sweep map was defined by Armstronget al. (2014b). The concept of the sweep map is as follows. Given a path one assigns to each step a label,the labels being distinct integers. To obtain the image of a path under the sweep map, one rearranges thesteps such that the labels are in increasing order.We now give a description of the zeta map of type C similar to the sweep map on Dyck paths. Given apath π = s s , . . . , s n ∈ L n,n assign a label to each step by setting ℓ ( s ) = 0 , ℓ ( s i +1 ) = ℓ ( s i ) + 2 n + 1 if s i = N , and ℓ ( s i +1 ) = ℓ ( s i ) − n if s i = E . Now define a collection X of labelled steps as follows. If ℓ ( s i ) < then add ( s i , ℓ ( s i )) . If ℓ ( s i ) > then add ( s i − , − ℓ ( s i )) . Finally, for the step s which is theonly step labelled , add ( s n , − n ) . Thus, X contains n labelled steps.Finally, draw a path as follows. Choose ( s, ℓ ) ∈ X such that ℓ is the minimal label among all pairs in X . Draw the step s and remove ( s, ℓ ) from X . Repeat until X is empty. We denote the path obtained inthis way by sw( π ) . See Figure 14. We conclude with the following theorem. Theorem 4.4
For each lattice path π ∈ L n,n we have sw( π ) = ζ ( π ) . In particular, the sweep map sw : L n,n → B n is a bijection. References
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