Generalized Dyck tilings (Extended Abstract)

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Generalized Dyck tilings (Extended Abstract)

Matthieu Josuat-Vergès, Jang Soo Kim

To cite this version:

Matthieu Josuat-Vergès, Jang Soo Kim. Generalized Dyck tilings (Extended Abstract). 26th Interna-

tional Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago,

United States. pp.181-192. �hal-01058407v2�

FPSAC 2014, Chicago, USA DMTCS proc.AT, 2014, 181–192

Generalized Dyck tilings (Extended Abstract)

Matthieu Josuat-Verg´es

and Jang Soo Kim

1IGM, Universit´e de Marne-la-Vall´ee, France

2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea.

Abstract.Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations:k-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analogue of the Bruhat order on them. We show that certain families ofk-Dyck tilings are in bijection with intervals in this order.

We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.

R´esum´e.R´ecemment, Kenyon et Wilson ont introduit les pavages de Dyck, qui sont des pavages de la r´egion comprise entre deux chemins de Dyck. L’´enum´eration des pavages de Dyck est reli´ee aux formules d’´equerre sur les forˆets et `a la combinatoire des polynˆomes de Hermite. Le premier but de ce travail est de donner un point de vue alternatif sur les pavages de Dyck, en utilisant l’ordre faible et l’ordre de Bruhat sur les permutations. Nous introduisons ensuite deux g´en´eralisations naturelles: lesk-pavages de Dyck et les pavages de Dyck sym´etriques. Nous sommes amen´es `a consid´erer les permutations de Stirling, et d´efinissons un analogue de l’ordre de Bruhat. Nous montrons que certaines familles dek-pavages de Dyck sont en bijection avec des intervalles de cet ordre. Nous ´enum´erons les pavages de Dyck sym´etriques et montrons que certaines familles de pavages de Dyck sym´etriques sont en bijection avec des intervalles de l’ordre faible sur les permutations sign´ees.

Keywords:Dyck tilings, Bruhat order, weak order, Stirling permutations, forests, matchings

1 Introduction

Dyck tilings were recently introduced by Kenyon and Wilson [9] in the study of probabilities of statis- tical physics model called “double-dimer model”, and independently by Shigechi and Zinn-Justin [13]

in the study of Kazhdan-Lusztig polynomials. Dyck tilings also have connection with fully packed loop configurations [6] and the representation of symmetric groups [5].

The main purpose of this paper is to give a new point of view on Dyck tilings in terms of the weak order and the Bruhat order on permutations and to consider two natural generalizations of Dyck tilings.

∗Email:matthieu.josuat-verges@univ-mlv.fr

†Email:jangsookim@skku.edu. The second author was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2061006).

1365–8050 c2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

2 Matthieu Josuat-Verg`es and Jang Soo Kim

Fig. 1:An example of Dyck tiling.

ADyck path of length2nis a lattice path consisting of up steps(0,1)and down steps(1,0)from the origin(0,0)to the point(n, n)which never goes below the liney=x. We will also consider a Dyck path λof length2nas the Young diagram whose border is determined byλand the linesx= 0andy=n.

Suppose that λandµare Dyck paths of length2nwithµweakly aboveλ. ADyck tileis a ribbon whose northwest boundary is a Dyck path. ADyck tilingofλ/µis a tilingD of the region betweenλ andµwith Dyck tiles satisfying thecover-inclusive property: ifηis a tile ofD, then the translation ofη by(1,−1)is either completely belowλor contained in another tile ofD. See Figure 1 for an example.

We denote byD(λ/µ)the set of Dyck tilings ofλ/µ. ForD ∈ D(λ/µ)we callλandµthelower path and theupper pathofD, respectively. Then the set of Dyck tilings with fixed upper pathλis denoted by D(λ/∗)and similarly, the set of Dyck tilings with fixed lower pathµis denoted byD(∗/µ).

ForD ∈ D(λ/µ)we have two natural statistics: the areaarea(D)of the regionλ/µand the number tiles(D)of tiles ofD. We also consider the statisticart(D) = (area(D) + tiles(D))/2.

Kenyon and Wilson [9] conjectured the following two formulas:

!

D∈D(λ/∗)

q^art(D)= [n]q!

"

x∈F[hx]q

, (1)

!

D∈D(∗/µ)

q^tiles(D)= #

u∈UP(µ)

[ht(u)]q, (2)

whereF is the plane forest corresponding toλand, for a vertexx ∈F,hxdenotes the hook length of x. The set of up steps of a Dyck pathµis denoted byUP(µ)and foru∈ UP(µ),ht(u)is the number of squares between uand the line y = xplus 1. Here we use the standard notation forq-factorials:

[n]q = 1 +q+q²+· · ·+qⁿ⁻¹and[n]q! = [1]q[2]q. . .[n]q.

Formula (1) was first proved by Kim [10] non-bijectively and then by Kim, M´esz´aros, Panova, and Wilson [11] bijectively. In [11], they find a bijection betweenD(λ/∗)and increasing labelings of the plane forest corresponding toλ. Kim [10] and Konvalinka independently proved (2) by finding a bijection betweenD(∗/µ)and certain labelings ofµcalled Hermite histories.

In this paper we first show that, using the results of Bj¨orner and Wachs [2, 3], (1) can be interpreted as the length generating function for permutationsπ ≥^L σ in the left weak order for a 213-avoiding permutationσ. We also show that (2) is the length generating function for permutationsπ ≥ σin the Bruhat order for a 132-avoiding permutationσ. We then consider two natural generalizations of Dyck tilings, namely,k-Dyck tilings and symmetric Dyck tilings.

Fig. 1:An example of Dyck tiling.

ADyck path of length2nis a lattice path consisting of up steps(0,1)and down steps(1,0)from the origin(0,0)to the point(n, n)which never goes below the liney=x. We will also consider a Dyck path λof length2nas the Young diagram whose border is determined byλand the linesx= 0andy=n.

Suppose thatλandµ are Dyck paths of length2nwithµweakly aboveλ. ADyck tileis a ribbon whose northwest boundary is a Dyck path. ADyck tilingofλ/µis a tilingD of the region betweenλ andµwith Dyck tiles satisfying thecover-inclusive property: ifηis a tile ofD, then the translation ofη by(1,−1)is either completely belowλor contained in another tile ofD. See Figure 1 for an example.

We denote byD(λ/µ)the set of Dyck tilings ofλ/µ. ForD ∈ D(λ/µ)we callλandµthelower path and theupper pathofD, respectively. Then the set of Dyck tilings with fixed upper pathλis denoted by D(λ/∗)and similarly, the set of Dyck tilings with fixed lower pathµis denoted byD(∗/µ).

ForD ∈ D(λ/µ)we have two natural statistics: the areaarea(D)of the regionλ/µand the number tiles(D)of tiles ofD. We also consider the statisticart(D) = (area(D) + tiles(D))/2.

Kenyon and Wilson [9] conjectured the following two formulas:

X

D∈D(λ/∗)

q^art(D)= [n]q! Q

x∈F[hx]q

, (1)

X

D∈D(∗/µ)

q^tiles(D)= Y

u∈UP(µ)

[ht(u)]q, (2)

whereF is the plane forest corresponding toλand, for a vertexx∈ F,hxdenotes the hook length of x. The set of up steps of a Dyck pathµis denoted byUP(µ)and foru∈UP(µ),ht(u)is the number of squares betweenuand the line y = xplus 1. Here we use the standard notation for q-factorials:

[n]q = 1 +q+q²+· · ·+qⁿ⁻¹and[n]q! = [1]q[2]q. . .[n]q.

Formula (1) was first proved by Kim [10] non-bijectively and then by Kim, M´esz´aros, Panova, and Wilson [11] bijectively. In [11], they find a bijection betweenD(λ/∗)and increasing labelings of the plane forest corresponding toλ. Kim [10] and Konvalinka independently proved (2) by finding a bijection betweenD(∗/µ)and certain labelings ofµcalled Hermite histories.

In this paper we first show that, using the results of Bj¨orner and Wachs [2, 3], (1) can be interpreted as the length generating function for permutations π ≥^L σin the left weak order for a 213-avoiding permutationσ. We also show that (2) is the length generating function for permutationsπ ≥ σin the Bruhat order for a 132-avoiding permutationσ. We then consider two natural generalizations of Dyck tilings, namely,k-Dyck tilings and symmetric Dyck tilings.

Generalized Dyck tilings (Extended Abstract) 183

Generalized Dyck tilings 3

×× ×××

"→

× ××× ×

Fig. 2:The bijection from 132-avoiding permutations to Dyck paths. The crosses represent inversions of the permu- tation34215.

"→

Fig. 3:The bijection from Dyck paths to plane forests. Here the Dyck path is rotated clockwise by an angle of45^◦. A pair of matching up step and down step corresponds to a vertex.

The first generalization isk-Dyck tiling, where we usek-Dyck paths andk-Dyck tiles with the same cover-inclusive property. We generalize (2) by finding a bijection betweenk-Dyck tilings andk-Hermite histories. We considerk-Stirling permutations introduced by Gessel and Stanley [7]. We define a k- Bruhat order onk-Stirling permutations and show thatk-Dyck tilings with fixed upper path are in bijection with an interval in this order. We also consider a connection withk-regular noncrossing partitions. We generalize (1) tok-Dyck tilings with fixed lower pathλwhenλis a zigzag path.

The second generalization is symmetric Dyck tiling, which is invariant under the reflection along a line.

We show that symmetric Dyck tilings are in bijection with symmetric matchings and “marked” increasing labelings of symmetric forests. In a special case we show that symmetric Dyck tilings with fixed lower path correspond to an interval of the left weak order on signed permutations.

2 Dyck tilings as intervals of the Bruhat order and weak order

As we have seen in the introduction, the two natural points of view for enumerating Dyck tilings are when we fix the upper path, and when we fix the lower path. We show in this section that both can be interpreted in terms of permutations, using respectively the Bruhat order and the (left) weak order, see [1].

We begin by the case of a fixed upper path, and the Bruhat order. We need a bijectionαbetween 132-avoiding permutations and Dyck paths. This simple bijection is illustrated in Figure 2.

We denote bySn the set of permutations of[n] :={1,2, . . . , n}. Aninversionofπ∈ Snis a pair (i, j)of integers1 ≤i < j ≤nsuch thatπ(i) > π(j). The number of inversions ofπis denoted by inv(π). For a permutationτ, the set ofτ-avoiding permutations inSnis denoted bySn(τ).

Proposition 2.1 Suppose that a Dyck pathµcorresponds toσ∈Sn(132), then

!

D∈D(∗/µ)

q^tiles(D)=!

π≥σ

q^{inv(π)−inv(σ)},

whereπ≥σis the Bruhat order onSn.

Let us turn to the case of a fixed lower path in Dyck tilings. We can identify a Dyck pathλwith a plane forest as shown in Figure 3. For an increasing labelingLof a plane forestF letpost(L)denote the permutation obtained by readingLfrom right to left using post-order. See Figure 4. It is easy to see

7→

Generalized Dyck tilings 3

×× ×××

"→

× ××× ×

Fig. 2:The bijection from 132-avoiding permutations to Dyck paths. The crosses represent inversions of the permu- tation34215.

"→

Fig. 3:The bijection from Dyck paths to plane forests. Here the Dyck path is rotated clockwise by an angle of45^◦. A pair of matching up step and down step corresponds to a vertex.

The first generalization isk-Dyck tiling, where we usek-Dyck paths andk-Dyck tiles with the same cover-inclusive property. We generalize (2) by finding a bijection betweenk-Dyck tilings andk-Hermite histories. We consider k-Stirling permutations introduced by Gessel and Stanley [7]. We define a k- Bruhat order onk-Stirling permutations and show thatk-Dyck tilings with fixed upper path are in bijection with an interval in this order. We also consider a connection withk-regular noncrossing partitions. We generalize (1) tok-Dyck tilings with fixed lower pathλwhenλis a zigzag path.

The second generalization is symmetric Dyck tiling, which is invariant under the reflection along a line.

We show that symmetric Dyck tilings are in bijection with symmetric matchings and “marked” increasing labelings of symmetric forests. In a special case we show that symmetric Dyck tilings with fixed lower path correspond to an interval of the left weak order on signed permutations.

2 Dyck tilings as intervals of the Bruhat order and weak order

As we have seen in the introduction, the two natural points of view for enumerating Dyck tilings are when we fix the upper path, and when we fix the lower path. We show in this section that both can be interpreted in terms of permutations, using respectively the Bruhat order and the (left) weak order, see [1].

We begin by the case of a fixed upper path, and the Bruhat order. We need a bijectionαbetween 132-avoiding permutations and Dyck paths. This simple bijection is illustrated in Figure 2.

We denote bySn the set of permutations of[n] :={1,2, . . . , n}. Aninversionofπ ∈ Sn is a pair (i, j)of integers1 ≤i < j ≤nsuch thatπ(i) > π(j). The number of inversions ofπis denoted by inv(π). For a permutationτ, the set ofτ-avoiding permutations inSnis denoted bySn(τ).

Proposition 2.1 Suppose that a Dyck pathµcorresponds toσ∈Sn(132), then

!

D∈D(∗/µ)

q^tiles(D)=!

π≥σ

q^{inv(π)−inv(σ)},

whereπ≥σis the Bruhat order onSn.

Let us turn to the case of a fixed lower path in Dyck tilings. We can identify a Dyck pathλwith a plane forest as shown in Figure 3. For an increasing labelingLof a plane forestF letpost(L)denote the permutation obtained by readingLfrom right to left using post-order. See Figure 4. It is easy to see Fig. 2:The bijection from 132-avoiding permutations to Dyck paths. The crosses represent inversions of the permu- tation34215.

Generalized Dyck tilings 3

×× ×××

"→

× ××× ×

Fig. 2:The bijection from 132-avoiding permutations to Dyck paths. The crosses represent inversions of the permu- tation34215.

"→

Fig. 3:The bijection from Dyck paths to plane forests. Here the Dyck path is rotated clockwise by an angle of45^◦. A pair of matching up step and down step corresponds to a vertex.

labelings of symmetric forests. In a special case we show that symmetric Dyck tilings with fixed lower path correspond to an interval of the left weak order on signed permutations.

2 Dyck tilings as intervals of the Bruhat order and weak order

As we have seen in the introduction, the two natural points of view for enumerating Dyck tilings are when we fix the upper path, and when we fix the lower path. We show in this section that both can be interpreted in terms of permutations, using respectively the Bruhat order and the (left) weak order, see [1].

We begin by the case of a fixed upper path, and the Bruhat order. We need a bijectionαbetween 132-avoiding permutations and Dyck paths. This simple bijection is illustrated in Figure 2.

We denote bySn the set of permutations of[n] :={1,2, . . . , n}. Aninversionofπ ∈ Sn is a pair (i, j)of integers1 ≤i < j ≤nsuch thatπ(i) > π(j). The number of inversions ofπis denoted by inv(π). For a permutationτ, the set ofτ-avoiding permutations inSnis denoted bySn(τ).

Proposition 2.1 Suppose that a Dyck pathµcorresponds toσ∈Sn(132), then

!

D∈D(∗/µ)

q^tiles(D)=!

π≥σ

q^{inv(π)−inv(σ)},

whereπ≥σis the Bruhat order onSn.

Let us turn to the case of a fixed lower path in Dyck tilings. We can identify a Dyck pathλwith a plane forest as shown in Figure 3. For an increasing labelingLof a plane forestF letpost(L)denote the permutation obtained by readingLfrom right to left using post-order. See Figure 4. It is easy to see that there is a unique increasing labelingL0ofF such thatinv(post(L0))is minimal. In this case the permutationπ0= post(L0)is 213-avoiding.

Proposition 2.2 Letλbe a Dyck path with corresponding plane forestF. LetL0be the unique increasing labelingL0ofFsuch thatinv(post(L0))is minimal, andπ0= post(L0). Then

!

D∈D(λ/∗)

q^art(D)= !

π≥Lπ0

q^{inv(π)−inv(π0)}, where≥^Lis the left weak order onSn.

7→

Generalized Dyck tilings 3

×× ×××

"→

× ××× ×

Fig. 2:The bijection from 132-avoiding permutations to Dyck paths. The crosses represent inversions of the permu- tation34215.

"→

Fig. 3:The bijection from Dyck paths to plane forests. Here the Dyck path is rotated clockwise by an angle of45^◦. A pair of matching up step and down step corresponds to a vertex.

labelings of symmetric forests. In a special case we show that symmetric Dyck tilings with fixed lower path correspond to an interval of the left weak order on signed permutations.

2 Dyck tilings as intervals of the Bruhat order and weak order

As we have seen in the introduction, the two natural points of view for enumerating Dyck tilings are when we fix the upper path, and when we fix the lower path. We show in this section that both can be interpreted in terms of permutations, using respectively the Bruhat order and the (left) weak order, see [1].

We begin by the case of a fixed upper path, and the Bruhat order. We need a bijectionαbetween 132-avoiding permutations and Dyck paths. This simple bijection is illustrated in Figure 2.

We denote bySnthe set of permutations of[n] := {1,2, . . . , n}. Aninversionofπ ∈ Sn is a pair (i, j)of integers1 ≤ i < j ≤ nsuch thatπ(i)> π(j). The number of inversions ofπis denoted by inv(π). For a permutationτ, the set ofτ-avoiding permutations inSnis denoted bySn(τ).

Proposition 2.1 Suppose that a Dyck pathµcorresponds toσ∈Sn(132), then

!

D∈D(∗/µ)

q^tiles(D)=!

π≥σ

qinv(π)−inv(σ),

whereπ≥σis the Bruhat order onSn.

Let us turn to the case of a fixed lower path in Dyck tilings. We can identify a Dyck pathλwith a plane forest as shown in Figure 3. For an increasing labelingLof a plane forestF letpost(L)denote the permutation obtained by readingLfrom right to left using post-order. See Figure 4. It is easy to see that there is a unique increasing labelingL0 ofF such thatinv(post(L0))is minimal. In this case the permutationπ0= post(L0)is 213-avoiding.

Proposition 2.2 Letλbe a Dyck path with corresponding plane forestF. LetL0be the unique increasing labelingL0ofFsuch thatinv(post(L0))is minimal, andπ0= post(L0). Then

!

D∈D(λ/∗)

q^art(D)= !

π≥Lπ0

q^{inv(π)−inv(π0)}, where≥^Lis the left weak order onSn.

Fig. 3:The bijection from Dyck paths to plane forests. Here the Dyck path is rotated clockwise by an angle of45^◦. A pair of matching up step and down step corresponds to a vertex.

The first generalization isk-Dyck tiling, where we usek-Dyck paths andk-Dyck tiles with the same cover-inclusive property. We generalize (2) by finding a bijection betweenk-Dyck tilings andk-Hermite histories. We considerk-Stirling permutations introduced by Gessel and Stanley [7]. We define a k- Bruhat order onk-Stirling permutations and show thatk-Dyck tilings with fixed upper path are in bijection with an interval in this order. We also consider a connection withk-regular noncrossing partitions. We generalize (1) tok-Dyck tilings with fixed lower pathλwhenλis a zigzag path.

The second generalization is symmetric Dyck tiling, which is invariant under the reflection along a line.

We show that symmetric Dyck tilings are in bijection with symmetric matchings and “marked” increasing labelings of symmetric forests. In a special case we show that symmetric Dyck tilings with fixed lower path correspond to an interval of the left weak order on signed permutations.

2 Dyck tilings as intervals of the Bruhat order and weak order

As we have seen in the introduction, the two natural points of view for enumerating Dyck tilings are when we fix the upper path, and when we fix the lower path. We show in this section that both can be interpreted in terms of permutations, using respectively the Bruhat order and the (left) weak order, see [1].

We begin by the case of a fixed upper path, and the Bruhat order. We need a bijection αbetween 132-avoiding permutations and Dyck paths. This simple bijection is illustrated in Figure 2.

We denote bySn the set of permutations of[n] := {1,2, . . . , n}. Aninversionofπ ∈Sn is a pair (i, j)of integers1 ≤ i < j ≤ nsuch thatπ(i) > π(j). The number of inversions ofπis denoted by inv(π). For a permutationτ, the set ofτ-avoiding permutations inSnis denoted bySn(τ).

Proposition 2.1 Suppose that a Dyck pathµcorresponds toσ∈Sn(132), then X

D∈D(∗/µ)

q^tiles(D)=X

π≥σ

qinv(π)−inv(σ),

whereπ≥σis the Bruhat order onSn.

Let us turn to the case of a fixed lower path in Dyck tilings. We can identify a Dyck pathλwith a plane forest as shown in Figure 3. For an increasing labelingLof a plane forestF letpost(L)denote

4 Matthieu Josuat-Verg`es and Jang Soo Kim 13

3 8

1 7

2

9 10

5 6

4 12

11

Fig. 4:An increasing labelingLof a forest of size 13. We havepost(L) = 11,12,6,10,5,4,9,2,7,8,13,3,1.

Fig. 5:An example of2-Dyck tiling.

Note that the inversion generating function of increasing labelings of a plane forest is given by a hook length formula [2]. The fact that some intervals for the weak order have a generating function given by a hook length formula follows from [3].

3 k-Dyck tilings

Ak-Dyck pathis a lattice path consisting ofup steps(0,1)anddown steps(1,0)from the origin(0,0) to the point(kn, n)which never goes below the liney=x/k. LetDyck^(k)(n)denote the set ofk-Dyck paths from(0,0)to(kn, n).

A k-Dyck tileis a ribbon in which the centers of the cell form ak-Dyck path. A(cover-inclusive) k-Dyck tilingis a tilingDof the regionλ/µbetweenλ, µ∈Dyck^(k)(n)withk-Dyck tiles satisfying the cover-inclusive property: ifηis a tile inD, then the translation ofηby(1,−1)is either completely below λor contained in another tile ofD. See Figure 5 for an example. We denote byD^(k)(λ/µ)the set of k-Dyck tilings ofλ/µ. We also denote byD^(k)(λ/∗)andD^(k)(∗/µ)the sets ofk-Dyck tilings with fixed lower pathλand with fixed upper pathµ, respectively.

For an up stepuofµ∈Dyck^(k)(n), we defineht(u)to be the number of squares betweenuand the liney=x/kplus 1. ForD ∈ D^(k)(λ/µ), there are two natural statistics: the numbertiles(D)of tiles in Dand the areaarea(D)of the region occupied byD. We also define

artk(D) =k·area(D) + tiles(D)

k+ 1 .

The following theorem is a generalization of (2).

Theorem 3.1 Forµ∈Dyck^(k)(n), we have

!

D∈D^(k)(∗/µ)

q^tiles(D)= "

u∈UP(µ)

[ht(u)]_q. (3)

Fig. 4:An increasing labelingLof a forest of size 13. We havepost(L) = 11,12,6,10,5,4,9,2,7,8,13,3,1.

the permutation obtained by readingLfrom right to left using post-order. See Figure 4. It is easy to see that there is a unique increasing labelingL0 ofF such thatinv(post(L0))is minimal. In this case the permutationπ0= post(L0)is 213-avoiding.

Proposition 2.2 Letλbe a Dyck path with corresponding plane forestF. LetL0be the unique increasing labelingL0ofFsuch thatinv(post(L0))is minimal, andπ0= post(L0). Then

X

D∈D(λ/∗)

q^art(D)= X

π≥Lπ0

qinv(π)−inv(π0),

where≥^Lis the left weak order onSn.

Note that the inversion generating function of increasing labelings of a plane forest is given by a hook length formula [2]. The fact that some intervals for the weak order have a generating function given by a hook length formula follows from [3].

3 k-Dyck tilings

Ak-Dyck pathis a lattice path consisting ofup steps(0,1)anddown steps(1,0)from the origin(0,0) to the point(kn, n)which never goes below the liney =x/k. LetDyck^(k)(n)denote the set ofk-Dyck paths from(0,0)to(kn, n).

Ak-Dyck tileis a ribbon in which the centers of the cell form a k-Dyck path. A (cover-inclusive) k-Dyck tilingis a tilingDof the regionλ/µbetweenλ, µ∈Dyck^(k)(n)withk-Dyck tiles satisfying the cover-inclusive property: ifηis a tile inD, then the translation ofηby(1,−1)is either completely below λor contained in another tile ofD. See Figure 5 for an example. We denote byD^(k)(λ/µ)the set of k-Dyck tilings ofλ/µ. We also denote byD^(k)(λ/∗)andD^(k)(∗/µ)the sets ofk-Dyck tilings with fixed lower pathλand with fixed upper pathµ, respectively.

For an up stepuofµ∈Dyck^(k)(n), we defineht(u)to be the number of squares betweenuand the liney=x/kplus 1. ForD∈ D^(k)(λ/µ), there are two natural statistics: the numbertiles(D)of tiles in Dand the areaarea(D)of the region occupied byD. We also define

artk(D) = k·area(D) + tiles(D)

k+ 1 .

The following theorem is a generalization of (2).

Theorem 3.1 Forµ∈Dyck^(k)(n), we have X

D∈D^(k)(∗/µ)

q^tiles(D)= Y

u∈UP(µ)

[ht(u)]_q. (3)

Generalized Dyck tilings (Extended Abstract) 185

4 Matthieu Josuat-Verg`es and Jang Soo Kim

13

3 8

1 7

2

9 10

5 6

4 12

11

Fig. 4:An increasing labelingLof a forest of size 13. We havepost(L) = 11,12,6,10,5,4,9,2,7,8,13,3,1.

Fig. 5:An example of2-Dyck tiling.

Note that the inversion generating function of increasing labelings of a plane forest is given by a hook length formula [2]. The fact that some intervals for the weak order have a generating function given by a hook length formula follows from [3].

3 k-Dyck tilings

Ak-Dyck pathis a lattice path consisting ofup steps(0,1)anddown steps(1,0)from the origin(0,0) to the point(kn, n)which never goes below the liney =x/k. LetDyck^(k)(n)denote the set ofk-Dyck paths from(0,0)to(kn, n).

Ak-Dyck tileis a ribbon in which the centers of the cell form ak-Dyck path. A (cover-inclusive) k-Dyck tilingis a tilingDof the regionλ/µbetweenλ, µ∈Dyck^(k)(n)withk-Dyck tiles satisfying the cover-inclusive property: ifηis a tile inD, then the translation ofηby(1,−1)is either completely below λor contained in another tile ofD. See Figure 5 for an example. We denote byD^(k)(λ/µ)the set of k-Dyck tilings ofλ/µ. We also denote byD^(k)(λ/∗)andD^(k)(∗/µ)the sets ofk-Dyck tilings with fixed lower pathλand with fixed upper pathµ, respectively.

For an up stepuofµ∈Dyck^(k)(n), we defineht(u)to be the number of squares betweenuand the liney =x/kplus 1. ForD∈ D^(k)(λ/µ), there are two natural statistics: the numbertiles(D)of tiles in Dand the areaarea(D)of the region occupied byD. We also define

artk(D) =k·area(D) + tiles(D)

k+ 1 .

The following theorem is a generalization of (2).

Theorem 3.1 Forµ∈Dyck^(k)(n), we have

!

D∈D^(k)(∗/µ)

q^tiles(D)= "

u∈UP(µ)

[ht(u)]_q. (3)

Fig. 5:An example of2-Dyck tiling.

For an arbitraryλ ∈ Dyck^(k)(n), there does not seem to be a product formula for the cardinality of D^(k)(λ/∗). However, whenλis a zigzag path there is a nice generalization of (1).

Theorem 3.2 Letλbe the pathuⁿ¹d^kn1uⁿ²d^kn2· · ·u^{n`</sup>d<sup>kn`}, whereumeans an up step anddmeans a down step. Then we have

X

D∈D^(k)(λ/∗)

q^artk(D)=

kn1+n2

n2

q

k(n1+n2) +n3

n3

q· · ·

k(n1+· · ·+n<sub>`−1</sub>) +n`

n`

q

,

where_a

b

q= _[b−^[a]_a]q^q_![b]^! _q!.

4 k-Stirling permutations and the k-Bruhat order

In this section we considerk-Stirling permutations which were introduced by Gessel and Stanley [7] for k= 2and studied further for generalkby Park [12].

Ak-Stirling permutation of size nis a permutation of the multiset{1^k,2^k, . . . , n^k}such that if an integerjappears between twoi’s theni > j. LetS^(k)n denote the set ofk-Stirling permutations of sizen.

We can represent ak-Stirling permutationπ=π1π2. . . πknas then×knmatrixM = (Mi,j)defined byMi,j = 1ifπj =iand andMi,j = 0otherwise. Then the entries ofM are0’s and1’s such that each column contains exactly one1, each row containsk1’s, and it does not contain the following matrix as a

submatrix:

1 0 1 0 1 0

.

Ak-inversionofπ∈S^(k)n is a pair(i, j)∈[n]×[kn]such thatπj > iand the firstiappears afterπj. Equivalently we can think of ak-inversion as an entry in the matrix ofπwhich hask1’s to the left in the same row and one1below in the same column. We denote the set ofk-inversions ofπbyINVk(π), and invk(π) =|INVk(π)|. It is easy to see thatπis determined by itsk-inversions.

Definition 1 We define thek-Bruhat orderonS^(k)n given by the cover relationσlπifπis obtained from σby exchanging the two numbers in positionsa1and ak+1 for some integersa1 < a2 < · · · < ak+1

satisfying the following conditions:

1. σa1 =σa2 =· · ·=σak< σak+1, and

6 Matthieu Josuat-Verg`es and Jang Soo Kim 332211

332112 322311 331122 322113 223311 311322 321123 223113 113322 311223 221133

113223 211233 112233

Fig. 6:The Hasse diagram ofS⁽²⁾₃ .

××

"→

Fig. 7:The bijectionαin Proposition 4.3.

There is a trivial bijection betweenk-Dyck tilings without tiles, andk-Dyck paths. As for thek-Stirling permutations, we have:

Proposition 4.3 The inversions of a 132-avoidingk-Stirling permutationσare arranged as the cells of a top-left justified Ferrers diagram. Define a pathα(σ)from the bottom left to the top right corner, and following the boundary of this Ferrers diagram with up and right steps. Thenαis a bijection between 132-avoidingk-Stirling permutations andk-Dyck paths of lengthn, in particular both are counted by the Fuss-Catalan numbers_kn+1¹ !(k+1)n

n

"

.

The proof is simple and similar to the casek= 1. For example, see Figure 7.

We now have a generalization of Proposition 2.1.

Theorem 4.4 Ifµis ak-Dyck path corresponding toσ∈S^(k)n (132), then

#

D∈D^(k)n (∗/µ)

q^tiles(D)=#

π≥σ

q^{invk(π)−invk(σ)}.

5 k-regular noncrossing partitions

In this section, we take another point of view on thek-Stirling permutations studied in the previous section.

Definition 2 We denote byNC^(k)_n the set ofk-regular noncrossing partitions of sizen, i.e. set partitions of[kn]such that each block containskelements, and there are no integersa < b < c < dsuch thata,c are in one block, andb,din another block. To eachk-regular noncrossing partitionπof[kn], we define its nesting posetNest(π)as follows: the elements of the poset are the blocks ofπ, andx≤yin the poset when the blockxlies between two elements of the blocky.

Fig. 6:The Hasse diagram ofS⁽²⁾₃ .

6 Matthieu Josuat-Verg`es and Jang Soo Kim

332211 332112 322311 331122 322113 223311 311322 321123 223113 113322 311223 221133

113223 211233 112233

Fig. 6:The Hasse diagram ofS⁽²⁾₃ .

××

"→

Fig. 7:The bijectionαin Proposition 4.3.

There is a trivial bijection betweenk-Dyck tilings without tiles, andk-Dyck paths. As for thek-Stirling permutations, we have:

Proposition 4.3 The inversions of a 132-avoidingk-Stirling permutationσare arranged as the cells of a top-left justified Ferrers diagram. Define a pathα(σ)from the bottom left to the top right corner, and following the boundary of this Ferrers diagram with up and right steps. Thenαis a bijection between 132-avoidingk-Stirling permutations andk-Dyck paths of lengthn, in particular both are counted by the Fuss-Catalan numbers_kn+1¹ !(k+1)n

n

"

.

The proof is simple and similar to the casek= 1. For example, see Figure 7.

We now have a generalization of Proposition 2.1.

Theorem 4.4 Ifµis ak-Dyck path corresponding toσ∈S^(k)n (132), then

#

D∈Dn^(k)(∗/µ)

q^tiles(D)=#

π≥σ

q^{invk(π)−invk(σ)}.

5 k-regular noncrossing partitions

In this section, we take another point of view on thek-Stirling permutations studied in the previous section.

Definition 2 We denote byNC^(k)_n the set ofk-regular noncrossing partitions of sizen, i.e. set partitions of[kn]such that each block containskelements, and there are no integersa < b < c < dsuch thata,c are in one block, andb,din another block. To eachk-regular noncrossing partitionπof[kn], we define its nesting posetNest(π)as follows: the elements of the poset are the blocks ofπ, andx≤yin the poset when the blockxlies between two elements of the blocky.

7→

6 Matthieu Josuat-Verg`es and Jang Soo Kim

332211 332112 322311 331122 322113 223311 311322 321123 223113 113322 311223 221133

113223 211233 112233

Fig. 6:The Hasse diagram ofS⁽²⁾₃ .

××

"→

Fig. 7:The bijectionαin Proposition 4.3.

There is a trivial bijection betweenk-Dyck tilings without tiles, andk-Dyck paths. As for thek-Stirling permutations, we have:

Proposition 4.3 The inversions of a 132-avoidingk-Stirling permutationσare arranged as the cells of a top-left justified Ferrers diagram. Define a pathα(σ)from the bottom left to the top right corner, and following the boundary of this Ferrers diagram with up and right steps. Then αis a bijection between 132-avoidingk-Stirling permutations andk-Dyck paths of lengthn, in particular both are counted by the Fuss-Catalan numbers_kn+1¹ !(k+1)n

n

"

.

The proof is simple and similar to the casek= 1. For example, see Figure 7.

We now have a generalization of Proposition 2.1.

Theorem 4.4 Ifµis ak-Dyck path corresponding toσ∈S^(k)n (132), then

#

D∈D^(k)n (∗/µ)

q^tiles(D)=#

π≥σ

q^{invk(π)−invk(σ)}.

5 k-regular noncrossing partitions

In this section, we take another point of view on thek-Stirling permutations studied in the previous section.

Definition 2 We denote byNC^(k)_n the set ofk-regular noncrossing partitions of sizen, i.e. set partitions of[kn]such that each block containskelements, and there are no integersa < b < c < dsuch thata,c are in one block, andb,din another block. To eachk-regular noncrossing partitionπof[kn], we define its nesting posetNest(π)as follows: the elements of the poset are the blocks ofπ, andx≤yin the poset when the blockxlies between two elements of the blocky.

Fig. 7:The bijectionαin Proposition 4.3.

2. for allak< i < ak+1we have eitherσi< σakorσi> σak+1.

Note that the1-Bruhat order is the usual Bruhat order. Figure 6 illustrates thek-Bruhat order.

Remark 1 The elements ofS^(k)n where all occurrences ofiare consecutive for anyi, form a subset in natural bijection with permutations. In general (k >1andn >2), the induced order on this subset is strictly contained in the Bruhat order, and contains strictly the left weak order.

Proposition 4.1 Thek-Bruhat order gives a graded posetS^(k)n with rank functioninvk.

Proposition 4.2 Suppose thatσ∈S^(k)n is132-avoiding. Then, forπ∈S^(k)n , we haveσ≤πif and only ifINVk(σ)⊆INVk(π).

There is a trivial bijection betweenk-Dyck tilings without tiles, andk-Dyck paths. As for thek-Stirling permutations, we have:

Proposition 4.3 The inversions of a 132-avoidingk-Stirling permutationσare arranged as the cells of a top-left justified Ferrers diagram. Define a pathα(σ)from the bottom left to the top right corner, and following the boundary of this Ferrers diagram with up and right steps. Thenαis a bijection between 132-avoidingk-Stirling permutations andk-Dyck paths of lengthn, in particular both are counted by the Fuss-Catalan numbers _kn+1^{1 (k+1)n}_n

.

The proof is simple and similar to the casek= 1. For example, see Figure 7.

We now have a generalization of Proposition 2.1.

Generalized Dyck tilings (Extended Abstract) 187 Theorem 4.4 Ifµis ak-Dyck path corresponding toσ∈S^(k)n (132), then

X

D∈D^(k)n (∗/µ)

q^tiles(D)=X

π≥σ

q^{invk(π)−invk(σ)}.

5 k-regular noncrossing partitions

In this section, we take another point of view on thek-Stirling permutations studied in the previous section.

Definition 2 We denote byNC^(k)_n the set ofk-regular noncrossing partitions of sizen, i.e. set partitions of[kn]such that each block containskelements, and there are no integersa < b < c < dsuch thata,c are in one block, andb,din another block. To eachk-regular noncrossing partitionπof[kn], we define its nesting posetNest(π)as follows: the elements of the poset are the blocks ofπ, andx≤yin the poset when the blockxlies between two elements of the blocky.

There is a natural way to consider ak-Stirling permutation as a linear extension of the nesting poset of ak-regular noncrossing partition.

Proposition 5.1 There is a bijection betweenS^(k)n and the pairs(π, E)whereπ ∈ NC^(k)_n andE is a linear extension of the posetNest(π).

Note that since the nesting order of ak-noncrossing partition is a forest, there is a hook length formula for the number of linear extensions [3]. Ifx∈Nest(π), lethxdenote the number of elements abovexin the nesting poset, then the number of linear extensions ofπ∈NC^(k)_n is

Q n!

x∈Nest(π)hx

.

This gives the following formula for the number ofk-Stirling permutations.

Proposition 5.2 (Multifactorial hook length formula)

1(k+ 1)(2k+ 1). . .((n−1)k+ 1) = X

π∈NC^(k)n

Qn!

x∈Nest(π)

hx

.

In the casek >2, it is not clear how to follow theq-statistic, but fork = 2we can use the bijection between noncrossing matchings and plane forests (which is justπ7→Nest(π)), and use theq-hook length formula from [2]. We get a hook length formula for[1]q[3]q. . .[2n−1]q.

Proposition 5.3 (q-double factorial hook length formula) We have

[1]q[3]q. . .[2n−1]q =X

F

[n]q²!Y

v∈F

q^hv−1 [hv]q²

, where the sum is over all plane forestsFwithnvertices.

8 Matthieu Josuat-Verg`es and Jang Soo Kim

−5 8

9 −6

−1 17 10 13

−12

−7

−4

−2 16

−11 18

−14 15 3

Fig. 8:A symmetric plane tree of size18and a typeBincreasing labelingLof it. The center line is the dashed line.

We havepost(L) =−2,−4,8,−5,−6,−7,9,17,13,10,−1,−12,16,18,15,−11,3,−14.

2. for eachi∈[n], exactly one ofior−iappears, 3. the label of every center vertex is negative.

See Figure 8 for an example of typeB increasing labeling. We denote by the set of typeBincreasing labelings ofFbyINCB(F).

ForL∈INCB(F)we definepost(L)to be the signed permutation obtained by readingFfrom right to left using post-order. It is easy to see that there is a unique typeBincreasing labelingL0∈INCB(F) such thatℓB(post(L0))is minimal.

Forπ=π1π2. . . πn∈Bnwe defineNEG(π) ={i:πi<0},

INV(π) ={(i, j) : 1≤i < j ≤n, πi> πj},

andNEGPAR(π)to be the integer partition with part|πi|for eachπi<0. We will considerNEGPAR(π) as the set of the cells in the Young diagram.

The following proposition is well known, for instance, see [1, Proposition 8.1.1].

Proposition 6.1 Forπ∈Bnwe haveℓB(π) =|INV(π)|+|NEGPAR(π)|.

We have a simple criterion to determine whether two signed permutations are comparable in the left weak order≤^L.

Proposition 6.2 Forσ, π∈Bn, we haveσ≤^Lπif and only ifINV(σ)⊆INV(π)andNEGPAR(σ)⊆ NEGPAR(π). Moreover, in this case we haveNEG(σ)⊆NEG(π).

Proposition 6.3 LetF be a symmetric plane forest of sizen. LetL0be the unique typeB increasing labeling ofF withℓB(post(L0))minimal. Then

{post(L) :L∈INCB(F)}= [post(L0),¯1¯2. . .n],¯ where[σ, π]means the interval{τ∈Bn :σ≤^Lτ≤^Lπ}in the poset(Bn,≤^L).

Fig. 8:A symmetric plane tree of size18and a typeBincreasing labelingLof it. The center line is the dashed line.

We havepost(L) =−2,−4,8,−5,−6,−7,9,17,13,10,−1,−12,16,18,15,−11,3,−14.

6 Type B increasing forests and the weak order on B

Let [n] = {1,2, . . . , n} and [±n] = {1,2, . . . , n,−1,−2, . . . ,−n}. Asigned permutation is a per- mutationπof[±n]such thatπ(−i) = −π(i). We can representπas the sequenceπ1π2. . . πn where πi = π(i). LetBn denote the set of signed permutations of[±n]. We refer the reader to [1] for basic definitions and properties ofBnsuch as the length function`B, the Bruhat order and the weak order.

For simplicity we will write a negative integer−ias¯ito express a signed permutation.

Asymmetric plane forestis a plane forest which is invariant under the reflection about a line, called thecenter line. Acenter vertexis a vertex on the center line. The(weakly) left partof a symmetric plane forest is the subforest consisting of vertices on or to the left of the center line. Thesizeof a symmetric plane forest is the number of vertices in the left part of it.

LetFbe a symmetric plane forest of sizen. AtypeBincreasing labelingofFis a labeling of the left part ofF with[±n]such that

1. the labels are increasing from roots to leaves, 2. for eachi∈[n], exactly one ofior−iappears, 3. the label of every center vertex is negative.

See Figure 8 for an example of typeB increasing labeling. We denote by the set of typeB increasing labelings ofFbyINCB(F).

ForL∈INCB(F)we definepost(L)to be the signed permutation obtained by readingF from right to left using post-order. It is easy to see that there is a unique typeBincreasing labelingL0∈INCB(F) such that`B(post(L0))is minimal.

Forπ=π1π2. . . πn∈Bnwe defineNEG(π) ={i:πi<0},

INV(π) ={(i, j) : 1≤i < j≤n, πi> πj},

andNEGPAR(π)to be the integer partition with part|πi|for eachπi<0. We will considerNEGPAR(π) as the set of the cells in the Young diagram.

The following proposition is well known, for instance, see [1, Proposition 8.1.1].

Proposition 6.1 Forπ∈Bnwe have`B(π) =|INV(π)|+|NEGPAR(π)|.

Generalized Dyck tilings (Extended Abstract) 189

Generalized Dyck tilings 9

Fig. 9:An example of symmetric Dyck tiling. It is symmetric about the dashed line.

7 Symmetric Dyck tilings and marked increasing forests

LetF be a symmetric plane forest of sizen. Amarked increasing labelingofFis a labeling of the left part of F with[n]such that the labels are increasing from roots to leaves, each integer appears exactly once, and each non-center vertex may be marked with∗. We denote the set of marked increasing labelings ofFbyINC^∗(F). Note that in generalINC^∗(F)andINCB(F)have different cardinalities.

LetL∈INC^∗(F). We denote byMARK(L)the set of labels of the marked vertices inL. Aninversion ofLis a pair of vertices(u, v)such thatL(u)> L(v),uandvare incomparable, anduis to the left ofv.

LetINV(L)denote the set of inversions ofL.

A Dyck path λof length2nis calledsymmetric if it is invariant under the reflection along the line x+y=n. For two symmetric Dyck pathsλandµof length2n, a Dyck tiling ofλ/µis calledsymmetric if it is invariant under the reflection along the linex+y=n. See Figure 9 for an example of symmetric Dyck tiling. We denote byD^sym(λ/µ)the set of symmetric Dyck tilings of shapeλ/µ.

For a symmetric Dyck tilingD, apositive tileis a tile which lies strictly to the left of the center line, and azero tileis a tile which intersects with the center line. We denote bytiles+(D)andtiles0(D)the number of positive tiles and zero tiles, respectively. Note that the total number of tiles inDistiles(D) = 2·tiles+(D) + tiles0(D). We also definearea+(D)andarea0(D)to be the total area of positive tiles and zero tiles inD, respectively, and

art+(D) = area+(D) + tiles+(D)

2 , art0(D) =area0(D) + tiles0(D)

2 .

For example, if D is the symmetric Dyck tiling in Figure 9, we havetiles+(D) = 5,tiles0(D) = 3, area+(D) = 7,area0(D) = 19,art+(D) = (7 + 5)/2 = 6, andart0(D) = (19 + 3)/2 = 11.

Theorem 7.1 LetFbe a symmetric plane forest of sizenandλthe corresponding Dyck path. Then there is a bijectionφ: INC^∗(F)→ D^sym(λ/∗)such that ifφ(L) =D, thentiles0(D) =|MARK(L)|and

art+(D) + art0(D) =|INV(L)|+ !

i∈MARK(L)

(n+ 1−i).

Corollary 7.2 LetFbe a symmetric plane forest of sizenwithkcenter vertices andλthe corresponding Dyck path. Then

|D^sym(λ/∗)|= 2^n−k n!

"

x∈Thx

.

Fig. 9:An example of symmetric Dyck tiling. It is symmetric about the dashed line.

We have a simple criterion to determine whether two signed permutations are comparable in the left weak order≤^L.

Proposition 6.2 Forσ, π∈Bn, we haveσ≤^Lπif and only ifINV(σ)⊆INV(π)andNEGPAR(σ)⊆ NEGPAR(π). Moreover, in this case we haveNEG(σ)⊆NEG(π).

Proposition 6.3 Let F be a symmetric plane forest of sizen. LetL0 be the unique typeB increasing labeling ofFwith`B(post(L0))minimal. Then

{post(L) :L∈INCB(F)}= [post(L0),¯1¯2. . .¯n], where[σ, π]means the interval{τ ∈Bn:σ≤^Lτ≤^Lπ}in the poset(Bn,≤^L).

7 Symmetric Dyck tilings and marked increasing forests

LetF be a symmetric plane forest of sizen. Amarked increasing labelingofF is a labeling of the left part ofF with[n]such that the labels are increasing from roots to leaves, each integer appears exactly once, and each non-center vertex may be marked with∗. We denote the set of marked increasing labelings ofF byINC^∗(F). Note that in generalINC^∗(F)andINCB(F)have different cardinalities.

LetL∈INC^∗(F). We denote byMARK(L)the set of labels of the marked vertices inL. Aninversion ofLis a pair of vertices(u, v)such thatL(u)> L(v),uandvare incomparable, anduis to the left ofv.

LetINV(L)denote the set of inversions ofL.

A Dyck path λof length 2nis called symmetricif it is invariant under the reflection along the line x+y=n. For two symmetric Dyck pathsλandµof length2n, a Dyck tiling ofλ/µis calledsymmetric if it is invariant under the reflection along the linex+y =n. See Figure 9 for an example of symmetric Dyck tiling. We denote byD^sym(λ/µ)the set of symmetric Dyck tilings of shapeλ/µ.

For a symmetric Dyck tilingD, apositive tileis a tile which lies strictly to the left of the center line, and azero tileis a tile which intersects with the center line. We denote bytiles+(D)andtiles0(D)the number of positive tiles and zero tiles, respectively. Note that the total number of tiles inDistiles(D) = 2·tiles+(D) + tiles0(D). We also definearea+(D)andarea0(D)to be the total area of positive tiles and zero tiles inD, respectively, and

art+(D) =area+(D) + tiles+(D)

2 , art0(D) = area0(D) + tiles0(D)

2 .

For example, ifD is the symmetric Dyck tiling in Figure 9, we havetiles+(D) = 5, tiles0(D) = 3, area+(D) = 7,area0(D) = 19,art+(D) = (7 + 5)/2 = 6, andart0(D) = (19 + 3)/2 = 11.

Theorem 7.1 LetFbe a symmetric plane forest of sizenandλthe corresponding Dyck path. Then there is a bijectionφ: INC^∗(F)→ D^sym(λ/∗)such that ifφ(L) =D, thentiles0(D) =|MARK(L)|and

art+(D) + art0(D) =|INV(L)|+ X

i∈MARK(L)

(n+ 1−i).

Corollary 7.2 LetFbe a symmetric plane forest of sizenwithkcenter vertices andλthe corresponding Dyck path. Then

|D^sym(λ/∗)|= 2^n−k n!

Q

x∈Thx

. Ifk= 0, we have

X

D∈Dsym(λ/∗)

q^{art+(D)+art0(D)}t^tiles0(D)= (1 +tq)(1 +tq²)· · ·(1 +tqⁿ) [n]q! Q

x∈T[hx]q

.

As is already mentionedINCB(F)andINC^∗(F)have different cardinalities. However, ifF has no center vertice, then there is a simple bijection betweenINCB(F)andINC^∗(F).

Corollary 7.3 LetFbe a symmetric plane forest of sizenwith no center vertices andλthe corresponding Dyck path. LetL0be the unique typeBincreasing labeling ofFwith`B(post(L0))minimal andπ0 = post(L0). Then

X

D∈Dsym(λ/∗)

q^{art+(D)+art0(D)}t^tiles0(D)= X

π≥Lπ0

q^{`B(π)−`B(π0)}t^{|NEG(π)|−|NEG(π0)|}.

8 Symmetric Dyck tilings and symmetric Hermite histories

For a symmetric Dyck pathµof length2n, letµ⁺denote the subpath consisting of the firstnsteps. Note that each up step is matched with a unique down step in a Dyck path. An up step ofµ⁺is calledmatched if the corresponding down step inµlies inµ⁺andunmatchedotherwise.

Asymmetric Hermite historyis a symmetric Dyck pathµwith a labeling of the up steps ofµ⁺in such a way that every matched up step of heighthhas labeli∈ {0,1, . . . , h−1}and the labelsa1, a2, . . . , ak

of the unmatched up steps form an involutive sequence. Here, a sequence is calledinvolutiveif it can be obtained by the following inductive way.

• The empty sequence is defined to be involutive.

• The sequence (0) is the only involutive sequence of length 1.

• Fork ≥2, an involutive sequence of lengthkis either the sequence obtained from an involutive sequence of lengthk−1by adding 0 at the end or the sequence obtained from an involutive sequence of lengthk−2by adding an integerrat the end and inserting a0before the lastrintegers, including the newly added integer, for some1≤r≤k−1.

From the definition it is clear that the numbervkof involutive sequences of lengthksatisfies the recurrence vk =vk−1+(k−1)vk−2with initial conditionsv0=v1= 1. Thusvkis equal to the number of involutions inSn.

We denote byH^sym(µ)be the set of symmetric Hermite histories onµ. ForH∈ H^sym(µ), letkHkbe the sum of labels inHandpos(H)the number of positive labels on unmatched up steps.

Generalized Dyck tilings (Extended Abstract) 191 Proposition 8.1 There is a bijectionψ:H^sym(µ)→ D^sym(∗/µ)such that ifψ(H) =D, thenkHk = tiles+(D) + tiles0(D)andpos(H) = tiles0(D). Thus,

X

D∈Dsym(∗/µ)

q^{tiles+(D)+tiles0(D)}t^tiles0(D)= X

H∈Hsym(µ)

q^kHkt^pos(H).

Corollary 8.2 Letµbe a symmetric Dyck path such thatµ⁺haskunmatched up steps. Then X

D∈Dsym(∗/µ)

q^{tiles+(D)+tiles0(D)}t^tiles0(D)=fk(q, t) Y

u∈UP(µ⁺)

[ht(u)]q,

whereUP(µ⁺)is the set of matched up steps inµ⁺andfk(q, t)is defined byf0(q, t) =f1(q, t) = 1and fk(q, t) =fk−1(q, t) +tq[k−1]qfk−2(q, t)fork≥2.

Asymmetric matchingis a matching on[±n]such that if{i, j}is an arc, then{−i,−j}is also an arc.

We denote byM^sym(n)the set of symmetric matchings on[±n]. Note that symmetric matchings are in bijection with fixed-point-free involutions inBn.

LetM ∈ M^sym(n). Asymmetric crossingofM is a pair of arcs{a, b}and{c, d}satisfyinga < c <

b < dandb, d >0. A symmetric crossing({a, b},{c, d})is calledself-symmetricif{c, d}={−a,−b}. We denote bycr(M)andsscr(M)the number of symmetric crossings and self-symmetric crossings of M, respectively.

For a symmetric Dyck pathµof length2n, letM^sym(µ)denote the set of symmetric matchingsM on [±n]such that theith smallest vertex ofM is a left vertex of an arc if and only if theith step ofµis an up step.

Proposition 8.3 We have X

H∈Hsym(µ)

q^kHkt^pos(H)= X

M∈Msym(µ)

q^cr(M)t^sscr(M).

ForD ∈ D^sym(∗/µ), letht(D)denote the number of unmatched up steps inµ⁺. Using the result in [4] and [8] on a generating function for partial matchings we obtain the following formula.

Proposition 8.4 We have X

D∈Dsym(n)

q^{tiles+(D)+tiles0(D)}t^tiles0(D)s^ht(D)

= Xn

m=0

s^mfm(q, t) (1−q)^(n−m)/2

X

k≥0

n

n−k 2

− n

n−k 2 −1

(−1)^(k−m)/2q(^(k−m)/2+12 )k+m 2 k−m

2

q

,

wherefm(q, t)is defined in Corollary 8.2.

Ift =s = 0in the above proposition, then we get the generating function for the usual Dyck tilings according to the number of tiles.

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