Pattern distributions in Dyck paths with a rst return decomposition constrained by height

Pattern distributions in Dyck paths with a rst return decomposition constrained by height

Jean-Luc Baril

, Richard Genestier

, and Sergey Kirgizov

LIB, Univ. Bourgogne Franche-Comté, France

May 14, 2020

Abstract

We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a rst return decomposition constrained by height.

Keywords: Dyck and Motzkin paths, pattern statistic, distribution, popularity

1 Introduction and notation

Dyck paths with a constrained rst return decomposition were introduced in [4]

where the authors present both enumerative results using generating functions and a constructive bijection with the set of Motzkin paths. In [5], a similar study has been conducted for Motzkin, 2-colored Motzkin, Schröder and Riordan paths. In the literature, many papers deal with the enumeration of classical Dyck paths according to dierent parameters, e.g. length, number of peaks, valleys, double rises and other pattern occurrences [10,15,16,18,19,20,21,24].

Restricted classes of Dyck paths have also been considered, for instance Barcucci et al. [2] consider Dyck paths having a non-decreasing height sequence of valleys (see also [8,9]). Other papers deal with Motzkin paths using similar methods [3, 6,11,12,17,22,25]. Motzkin and Catalan numbers appear alongside in many situations [11] and several one-to-one correspondences exist between restricted Dyck paths and Motzkin paths. For instance, Dyck paths avoiding a triple rise are enumerated by the Motzkin numbers [7].

In this paper, we focus on the distribution and the popularity of patterns of length at most three in constrained Dyck paths dened in [4]. Our method consists in showing how patterns are getting transferred from constrained Dyck paths to Motzkin paths, which settles us in a more suitable ground in order to provide generating functions for the distribution and the popularity.

A Motzkin path of length n ¥ 0 is a lattice path consisting of at steps F p1,0q, up stepsU p1,1q and down stepsD p1,1q, starting at p0,0q,

∗Electronic address: [email protected]

†Electronic address: [email protected]

‡Electronic address: [email protected]; Corresponding author

ending at pn,0q and never going below the x-axis. For n ¥ 0, we denote by M_n the set of all Motzkin paths of length n and we set M ”

n¥0M_n. A Motzkin path of length2nwith no at steps is a Dyck path of semilengthn. For n¥0, letDn be the set of all Dyck paths of semilength nand D”

n¥0Dn. The cardinality ofDnis given by thenth Catalan numbercn _n¹₁ ²ⁿ_n

, which is the general term of the sequence A000108 in the On-line Encyclopedia of Integer Sequences of N.J.A. Sloane [23]. The cardinality ofMn is given by the nth Motzkin number °_tn{2u

k0 n 2k

c_k (seeA000108in [23]).

Any non-empty Dyck pathP PDhas a unique rst return decomposition [10]

of the form P U αDβ where α and β are two Dyck paths in D. In [4], the authors introduced the set D^h,¥ constituted of the empty Dyck path and the Dyck paths inDhaving a rst return decomposition satisfying

hpU αDq ¥hpβq

where α, β P D^h,¥ and h returns the maximal height of a Dyck path. For n ¥ 0, let D^h,_n^¥ be the subset of Dyck paths of semilength n in D^h,¥. For instance,D₃^h,¥ consists of four Dyck pathsU DU DU D,U U DDU D,U U DU DD andU U U DDD. In [4], the authors prove, using generating functions, thatD^h,_n^¥ andMn have the same cardinality, and they present also the following bijection φbetween these sets.

ForP PD^h,¥,

φpPq

$' '&

''

%

ifP ,

φpαqF ifP αU D, φpαqφpγqU φpβqD ifP αU U βDγD.

For instance, the images byφofU DU DU D,U U DDU D,U U DU DD,U U U DDD, U U U U DDDDU U U DDU DD are respectively F F F, U DF, F U D, U F D and U U DDF U F D. We refer to Figure 1for an illustration of this mapping.

α ÝÑ ^φpαq

α β γ

ÝÑ ^{φpαq φpγq φpβq} Figure 1: Illustration of the bijectionφbetweenD_n^h,¥ andM_n.

A statistic on a setS of paths is an association of an integer inZ to each path inS. For instance the map that returns the number of steps is a statistic, and we denote by 1 (resp. 0) the constant statistic that sends any path to 1 (resp. 0). LetS be the set of all statistics on a setS. For X,YPS, we dene the statistic X Y so that pX YqpPq XpPq YpPq for any P PS, which endows S with a Z-module structure. LetS and T be two sets of paths, and let S andT be the associated statistic sets. Two statistics X PS and Y P T have the same distribution if and only if there exists a bijectionf fromS toT

such that for anyP PS we haveXpPq YpfpPqq. In this case, we say that f transports the statisticXintoY, which can be shortly written with the statistic equationfpXq Y(or XYwheneverf is the identity).

A patternX of lengthk¥1occurs in a pathP if and only ifP containsX as a sequence of consecutive steps. Note that other variants of pattern denition exist in the literature (see for instance [1]). From a given patternX and a setS of paths, we associate the statisticXfromStoNsuch thatXpPq is the number of occurrences of the pattern X in P. The popularity of a patternX in S is the total number of occurrences of the pattern X in all paths P P S, that is

°

PPSXpPq.

For instance, if P U U U DDU DD then we have UDpPq UUpPq 2, DDDpPq 0. Moreover, ifS tU U DD, U DU Du then the popularity of the patternU DinSis3. Also, for any subsetSofM, we have the statistic equation UDsince for any path inS the number of up steps equals to the number of down steps. If we restrictStoD_n(resp. M_n), then we haveU D2n(resp.

U F Dn), wheren is the constant statisticP ÞÑn.

Considering the above bijectionφfrom D_n^h,¥ to M_n, the length ofφpPq is the semilength ofP and we easily deduce the statistic equation:

φpUq φpDq U D Fn. (1) In this paper, we present statistic equations showing how φ: D^h,_n^¥ Ñ M_n transports statistics associated to patterns of length at most three into linear combinations of other statistics. Then, this allows us to conduct our enumera- tive study in the more natural and simpler context of Motzkin paths, while a direct study on D_n^h,¥ is complicated due to the lack of an adequate recursive decomposition. So, we use statistic equations in order to derive bivariate gen- erating functions for the distribution of patterns of length at most three, and we deduce the pattern popularity thereafter. Section 2 deals with patterns of length 2 while Section 3 deals with patterns of length 3. Many of the resulting sequences correspond to existing entries in the On-line Encyclopedia of Integer Sequences of N.J.A. Sloane [23] enumerating a quantity of already known com- binatorial structures, and we also obtain new sequences not yet known in [23].

2 Patterns of length 2

In this part, we provide statistic equations showing how patterns of length two behave throughφ:D_n^h,¥ÑMn, which allows us to deduce generating functions for the distribution and the popularity of such patterns. See Table 1 for an illustration of the distributions and Table 2 for the rst terms of popularity sequences.

Theorem 1. For n¥0, the bijection φfromD^h,_n^¥ toMn transports statistics associated to patterns of length two as follows:

φpUDq F UD, (2)

φpUUq φpDDq U UU UF, (3)

φpDUq FF FU DF DU. (4)

Proof. For equation (2), we refer to [4].

For equation (3), we observe that for any Dyck path of semilengthnan up stepU is always followed byU or D, which implies the equality UU UDn onD^h,_n^¥. With a similar argument andUD, we haveF U UD UU UFn on Mn. Using eq. (2) we obtain φpUUq nφpUDq n F UD

U UU UF.

For equation (4), we observe that the statisticUU UD DU DDequals 2n1onD^h,_n^¥. Using the straightforward equalityUUDDonDn, we obtain DU2n12UUUD. Applying the bijectionφand using eqs. (2) and (3) we obtainφpDUq 2n12pU UU UFqpF UDq. On the other hand, on the set of Motzkin paths of lengthnwe have the statistic equationsU F Dn andFD FU FF DD DU DF UD UU UFn1, which induces 2n1U F D FD FU FF DD DU DF UD UU UF. Also, for any Motzkin path we haveUDwhich impliesUU UF UDUD FD DD, and thus UU UFFD DD. So, combining all these equations, we obtain φpDUq U F D FD FU FF DD DU DF UD UU UFU DUUUFFDDDFUDFF FU DF DU.

Theorem 2. The bivariate generating functions Fppx, yq where the coecient of xⁿy^k is the number of Dyck paths in D_n^h,¥ containing exactly k occurrences of the pattern pP tU D, U U, DD, DUu are

F_{U D}px, yq x²x²yxy 1b

4x² px²py1q xy1q²

2x² ,

F_{U U}px, yq F_DDpx, yq x²y²x²yx 1b

4x²y² px²ypy1q x 1q²

2x²y² ,

F_DUpx, yq x²yx²xy 1b

4x² px²py1q xy1q²

2x²y .

Proof. ForpU D, Corollary 3 in [4] provides directly the bivariate generating function as a solution of a functional equation on Motzkin paths with respect to the number of occurrences of patternsF andU D(see eq. (2)).

Forp U U, there are two ways to obtain the generating function. First, we know thatUU UDnfor Dyck paths of semilengthn. So, we can obtain directly the generating function by calculating FU Dpxy,¹_yq. Moreover we know that φpUUq U UU UF (see eq. (3)). So, we decompose the set M of all Motzkin paths in the following way:

M FM U DM UpMzqDM.

The generating function for FM is given by 1 xMpx, yq; the g.f.

for U DM is x²yMpx, yq since U D contains one occurrence ofU; the g.f. for UpMzqDMisx²y²pMpx, yq 1qMpx, yq since a path inUpMzqD starts with U and its rst two steps are either U U or U F. This induces the functional equation

Mpx, yq 1 xMpx, yq x²yMpx, yq x²y²pMpx, yq 1qMpx, yq,

which also gives the expected result.

ForpDU, the result is obtained by using the equalityDUUD1 and evaluating FDUpx, yq 1 ^{FU Dpx,yq}_y^{FU Dpx,0q}. Note that we can obtain this result using eq. (4).

Table1gives the number of paths inD_n^h,¥havingkpeaksU Dfor1¤n¤11 and1¤k¤8. Applying standard techniques from generating function theory, we verify that the second row corresponds to the sequence of quarter-squares numbers, tpn²{4qu, which is the third row of Lozani¢'s triangle (seeA034851and [14]). The third row is a shift of the sequenceA005994corresponding to alkane numbers lp7, nq from fth row of Lozani¢'s triangle, which enumerates certain symmetries exhibited by chemical entities (alkane) consisting of hydrogen and carbon atoms arranged in a tree-like structure. Third diagonal corresponds to the squares, while fourth diagonal generates octahedral numbers,np2n² 1q{3 (seeA005900).

kzn 1 2 3 4 5 6 7 8 9 10 11

1 1 1 1 1 1 1 1 1 1 1 1

2 1 2 4 6 9 12 16 20 25 30

3 1 3 9 19 38 66 110 170 255

4 1 4 16 44 111 240 485 900

5 1 5 25 85 260 676 1615

6 1 6 36 146 526 1602

7 1 7 49 231 959

8 1 8 64 344

9 1 . . . .

Σ 1 2 4 9 21 51 127 323 835 2188 5798

Table 1: Number of paths in D_n^h,¥ having k peaks U D, or equivalently k1 valleysDU, or equivalentlynkdouble risesU U.

Corollary 1. For n¥0, the popularity of pattern pP tU U, U D, DD, DUu in D^h,_n^¥ is given by the generating functionGppxq:

GU Dpxq px1q?

3x²2x 13x²2x 1 2xp3x1q , GU Upxq GDDpxq

?3x²2x 1 x² 2x2

x³3x²4x 2 2x²?

3x²2x 1 ,

GDUpxq x²1 ?

3x²2x 1x³3x²x 1 2x²?

3x²2x 1 .

Proof. The generating functionGppxq of popularity is directly deduced from the bivariate generating function of pattern distribution

G_ppxq BF_ppx, yq By

y1

.

See Table2for the rst terms of the popularity sequences. The popularity of the patternU Dgenerates a shift of the sequenceA025566in [23]. As suggested in [23], the same sequence enumerates the rst dierences of the directed ani- mals sequence A005773, and also Motzkin paths of length2nwhose last weak valley occurs immediately after stepn.

The popularity sequence for DU is the sequence A025567. As mentioned in [13] by Ferrari and Munarini, this sequence corresponds to the number of edges in Hasse diagram of Motzkin paths, where the partial order is dened by the coverings F F ÞÑ U D, F U ÞÑ U F, DF ÞÑ F D, DU ÞÑ F F. This is an immediate consequence of the fact that φ maps pattern DU from constrained Dyck paths to the patternsF F, F U, DF, DU in Motzkin paths.

The popularity sequence forU U (or DD) does not yet appear in [23].

Pattern Popularity sequence OEIS

UD 1,3,8,22,61,171,483,1373,3923,11257,32418,93644 A025566 DU 0,1,4,13,40,120,356,1050,3088,9069,26620,78133 A025567 UU, DD 0,1,4,14,44,135,406,1211,3592,10623,31260,92488

Table 2: Popularity of length two patterns inD^h,_n^¥ for1¤n¤12.

3 Patterns of length 3

In this part we investigate how φ transports statistics associated to patterns of length three, and for each of them we provide generating functions for the distribution and popularity.

We need the following notations. For a step X P tU, D, Fu, an occurrence of the pattern X inside a pathP is an occurrence of X^k, whereX^k consists of k consecutive repetitions of X, for some k ¥1. The associated statistic of X (denotedX ) will be equal to°

k¥1X^k whereX^k is the statistic giving the number of occurrences ofX^k. More generally, for two possibly empty sequences of stepsY and Z we dene the patternY X Z as patterns of the form Y X^kZ for k ¥ 1, and its associated statistic as YX Z °

k¥1YX^kZ. For instance, an occurrence of the pattern DU D can be an occurrence of DU D, DU U D, DU U U D, and so on... In the path U U FDU U DFDU DU U U DDF F DD we count three occurrences of the pattern DU D. A path contains a dotted pattern Y (resp. Y) if and only if it starts (resp. ends) withY, and we use the notations Y,Y for the associated statistics.

Any patternU U in a Dyck path is immediately followed by an up step or a down step, implying the statistic equation UU UUU UUD. Also, any pattern U U in a Dyck pathP is either at the beginning of P or immediately preceded by an up or a down step, soUU UU UUU DUU. Using similar arguments we obtain the two following systems of statistic equations.

paq

$' ''

&

'' '%

UUUUU UUD UUUUU DUU UU UDUUD DUD UD DUDUU DUD

, pbq

$' ''

&

'' '%

DDDDD UDD DDDDD DDU DD UDUDD UDU UD DUDDU UDU

.

Observe that for any P P D^h,_n^¥, n ¥ 1, we have UDpPq 1 (resp.

UUpPq 0) when P pU Dqⁿ, and UDpPq 0 (resp. UUpPq 1) oth- erwise. So, we have the statistic equations φpUDq δ_Fn and φpUUq 1φpUDq, where δ_Fn is the Dirac statistic dened byδ_FnpPq 1 whenever P Fⁿ, and0otherwise.

Knowing the image throughφof only one statisticXP tUUU,UUD,DUU,DUDu and using results from Section 2, we can obtain the expressions of the images of the three other statistics from the system paq. The same reasoning holds for the second system pbq. So, we split Section 3 into two subsections, each dealing with one of the two systems.

3.1 The patterns U U U, U U D, DU U, DU D

Theorem 3. Forn¥0, the bijectionφfromD_n^h,¥toMntransports the statistic UUDas follows:

φpUUDq UF D UD.

Proof. We proceed by induction on n. For n 1, we have UUDpU Dq 0. SinceφpU Dq F, we obtainUF DpFq UDpFq 0, and the result holds. We assume the result fork¤n, and we will prove it forn 1by distinguishing two main cases.

- WheneverP αU D, we haveφpPq φpαqF, and pUF D UDqpφpPqq pUF D UDqpφpαqq. Using induction hypothesis, pUF D UDqpφpαqq UUDpαq which is also equal to UUDpαU Dqq, proving the rst case.

- Whenever P αU U βDγD, we have φpPq φpαqφpγqU φpβqD, and pUF D UDqpφpPqq pUF D UDqpφpαqq pUF D UDqpφpγqU φpβqDq. Us- ing induction hypothesis, we obtain pUF D UDqpφpPqq UUDpαq pUF D UDqpφpγqU φpβqDq. Now we distinguish three subcases in order to prove that pUF D UDqpφpγqU φpβqDq UUDpU U βDγDq.

Ifβ is the empty path, then pUF D UDqpφpγqU φpβqDq 1 pUF D UDqpφpγqq 1 UUDpγq UUDpU U DγDq.

If β pU Dq^k for some k ¥ 1, then pUF D UDqpφpγqU φppU Dq^kqDq pUF D UDqpφpγqU F^kDq pUF D UDqpφpγqq 1 UUDpγq 1 UUDpU UpU Dq^kDγDq UUDpU U βDγDq.

Ifβ starts with a double rise U U, then φpβq contains at least one up step (φpβq F<sup>`</sup> for any ` ¥ 0). So, we have pUF D UDqpφpγqU φpβqDq pUF D UDqpφpγqq pUF D UDqpφpβqq UUDpγq UUDpβq, which equals toUUDpU U βDγDq.

Considering these three cases, the second case is proved and the induction is completed.

Theorem 4. Forn¥0, the bijectionφfromD_n^h,¥toM_ntransports the statistic UUUas follows:

φpUUUq UF D 2 UF U UU .

Proof. Using Theorem 3 and the equation UU UUD UUU of system paq, we have φpUUUq φpUUq φpUUDq φpUUq UF DUD, and using Theorem 1, we have φpUUUq U UU UFUF DUD. Since we have UUD UU UFonMn, we obtainφpUUUq 2pUU UFq UF D.Using the trivial equationUFUF D UF Ufor Motzkin paths, we complete the proof.

Theorem 5. Forn¥1, the bijectionφfromD_n^h,¥toMntransports the statistic DUUas follows:

φpDUUq UF D UD δFⁿ1,

where δ_Fn is the Dirac statistic dened by δ_FnpPq 1 wheneverP Fⁿ, and 0 otherwise.

Proof. Using the dierence of the rst two equations of paq, we obtainDUU UUDUU, and nally,φpDUUq φpUUDqφpUUq UF D UDφpUUq.

As discussed at the beginning of Section 3,φpUUq 1δ_Fn, which completes the proof.

Theorem 6. Forn¥1, the bijectionφfromD_n^h,¥toMntransports the statistic DUD as follows:

φpDUDq FUF DδFⁿ

Proof. We have DUD UDUUD UD, which implies that φpDUDq φpUDq φpUUDq φpUDq. From Theorem1, Theorem3and the observations after the systems (a) and (b) we obtainφpDUDq UD FUDUF DδFⁿ FUF Dδ_Fn.

Theorem 7. The bivariate generating functions F_ppx, yq where the coecient ofxⁿy^kis the number of Dyck paths inD^h,_n^¥containing exactlykoccurrences of the patternpP tU U U, U U D, DU U, DU Du are given by the following expressions listed in the same order as written above.

x³yx³x²y² 2x1 a

pxxy1qpx²xy x1qpx³x³y x²y² 2x²y2xy2x 1q

2x²y²px1q ,

x²y2x² 2x1 a

px²y1q px²y4x² 4x1q

2x²px1q ,

2xx²y1 a

px²y1q px²y4x² 4x1q

2x²ypx1q ,

x³yx³x²y² 2xy1 a

pxyx1qpx² yxx1qpx³yx³ x²y² 2x²y2xy2x 1q

2x²pxy1q .

Proof. For p U U U and using Theorem 4, we have φpUUUq UF D 2pUF U UUq. So, we decompose Motzkin paths according to the patterns U F D,U F U, andU Uin order to exhibit a functional equation havingFppx, yq as solution:

M FM U DM UM0DM UpMzpM0YqqDM,

whereM₀is the subset ofMconsisting of paths of the formF^kfork¥1. The generating function for FM U DMis1 xF_{U U U}px, yq x²F_{U U U}px, yq; the g.f. forUM0DMisx²y₁^x_xFU U Upx, yq since a path inUM0Dis an occurrence ofU F D; the g.f. forUpMzpM0YqqDMisx²y²

FU U Upx, yq ₁¹_x FU U Upx, yq since a path inUpMzpM0YqqDstarts with either an occurrence ofU U or an occurrence of U F U.

So the functional equation is:

F_{U U U}px, yq 1 xF_{U U U}px, yq x²F_{U U U}px, yq x²y x

1xF_{U U U}px, yq x²y²

F_{U U U}px, yq 1 1x

F_{U U U}px, yq. A simple calculation (with Maple for instance) provides the result.

All other generating functions are obtained using a similar method, so we do not give the proofs here.

The generating functionGppxq for the popularity of the patternpP tU U U, U U D, DU U, DU Du is obtained directly by evaluating ^BFp_B^p_y^x,yq

y1. Table 3 provides the rst terms of the associated sequences. See also Table4 for an illustration of the distribution ofpP tU U U, U U D, DU U, DU Du.

Pattern Popularity sequence OEIS

UUU 0,0,1,5,19,65,210,658,2023,6147,18534,55594

UUD 0,1,3,9,25,70,196,553,1569,4476,12826,36894 A097861 DUU 0,0,0,1,5,20,70,231,735,2289,7029,21384 A304011? DUD 0,1,4,12,35,100,286,819,2353,6780,19591,56749

Table 3: Popularity ofpP tU U U, U U D, DU U, DU Du inD^h,_n^¥for1¤n¤12. The popularity sequence forU U D inD^h,_n^¥ is the sequence A097861, corre- sponding to the number of humps in all Motzkin paths of length n (a hump equals U F DorU D in our notation). The popularity forDU U seems to gen- erate the sequenceA304011, but we did not succeed in proving this fact, so we leave it as a conjecture.

kzn 1 2 3 4 5 6 7 8 9 0 1 2 3 5 8 13 21 34 55 1 1 3 8 18 38 76 147 2 1 4 14 40 104 250

3 1 5 21 71 215

4 1 6 30 119

5 1 7 40

(a)U U U

kzn 1 2 3 4 5 6 7 8 9 0 1 1 1 2 3 6 10 20 36 1 1 2 3 7 13 30 58 130 2 1 3 6 16 35 91 199 3 1 4 10 30 75 216

4 1 5 15 50 140

5 1 6 21 77

(b)DU D kzn 1 2 3 4 5 6 7 8 9

0 1 1 1 1 1 1 1 1 1 1 1 3 7 15 31 63 127 255 2 1 5 18 56 160 432

3 1 7 34 138

4 1 9

(c)U U D

kzn 1 2 3 4 5 6 7 8 9 0 1 2 4 8 16 32 64 128 256 1 1 5 18 56 160 432

2 1 7 34 138

3 1 9

4

(d)DU U

Table 4: Number of paths from D_n^h,¥ having k occurrences of the considered pattern.

Dyck paths fromD^h,_n^¥avoidingU U U,DU Urespectively generate Fibonacci numbers and integer squares. Those avoiding DU D seem to correspond to A007562 (number of planted trees where non-root, non-leaf nodes at even dis- tance from root are of degree 2). At the present time, there is no closed form for the generating function of sequence A007562. Note that any path avoid- ing DU U has at most one occurrence of U U D. Also, Dyck paths containing two occurrences of U U D in D^h,_n^¥ generate a shift of sequenceA001793, which corresponds to a subsequence in the triangle of coecients of Chebyshev's poly- nomials which is sequenceA053120.

3.2 The patterns DDD, DDU, U DD, U DU

Theorem 8. Forn¥0, the bijectionφfromD_n^h,¥toMntransports the statistic UDUas follows:

φpUDUq FF FUD

Proof. We proceed by induction on n. For n 1, we have UDUpU Dq 0. WithφpU Dq F, we obtain FFpFq FUDpFq 0, and the result holds. We assume the result fork¤n, and we will prove it forn 1.

- WheneverP αU D, we haveφpPq φpαqF, and pFF FUDqpφpPqq FFpφpαqFq FUDpφpαqFq FFpφpαqFq FUDpφpαqq. We distinguish two cases: (i) φpαq ends with F, and (ii) otherwise. In the case (i), α ends with U D, and thus pFF FUDqpφpPqq 1 FFpφpαqq FUDpφpαqq. Using the in- duction hypothesis we have pFF FUDqpφpPqq 1 UDUpαq UDUpαU Dq UDUpPq. In the case (ii),αdoes not end withU D, and thusφpαq does not end

withF. So, we have pFF FUDqpφpPqq FFpφpαqq FUDpφpαqq, and using the induction hypothesis pFF FUDqpφpPqq UDUpαq UDUpαU Dq UDUpPq.

- Whenever P does not end with U D, we have P αU U βDγD, and pFF FUDqpφpPqq pFF FUDqpφpαqφpγqU φpβqDq. Note that α cannot end with U D, otherwise it would contradict P P D_n^h,¥. This means that φpαq cannot end with F. So, all the possible occurrences of F F in φpPq belong necessarily to φpαq, φpβq and φpγq, which implies that FFpφpPqq FFpφpαqq FFpφpβqq FFpφpγqq. On the other hand, the possible occurrences of F U Din φpPq belong necessarily toφpαq, φpβq,φpγq, and eventually at the junction of φpγq and U φpβqD whenever φpγq ends with F and φpβq . So, we distinguish two cases: (a)φpγq ends with F andβ , and pbq otherwise.

In the case (a), we haveFUDpφpPqq 1 FFpφpαqq FFpφpβqq FFpφpγqq FUDpφpαqq FUDpφpβqq FUDpφpγqq, and using the induction hypothesis, we obtainFUDpφpPqq 1 UDUpαq UDUpβq UDUpγq UDUpαU U βDγDq UDUpPq. In the case (b), we have FUDpφpPqq FFpφpαqq FFpφpβqq FFpφpγqq FUDpφpαqq FUDpφpβqq FUDpφpγqq, and using the induction hy-

pothesis, we obtainFUDpφpPqq UDUpαq UDUpβq UDUpγq UDUpαU U βDγDq UDUpPq. So, the proof is complete.

Theorem 9. Forn¥0, the bijectionφfromD_n^h,¥toMntransports the statistic UDD as follows:

φpUDDq FD UD FUU FUF.

Proof. Considering the third equation of system (b), we haveφpUDDq φpUDq φpUDUq φpUDq. Using Theorems 1 and 8, we obtain φpUDDq F UDFFFUD φpUDq. In any Motzkin path P, a at step is either at the end of P, or followed by F, or D, or U U, or U D, or U F, that is F F FF FD FUU FUD FUF. Then, we obtain φpUDDq F UD FD FUU FUFφpUDq. Using the denition ofφin Introduction, it is clear thatφtransportsUD intoF, and the result holds.

Theorem 10. For n ¥ 0, the bijection φ from D^h,_n^¥ to Mn transports the statisticDDU as follows:

φpDDUq DF DU FUU FUF.

Proof. Combining the fourth equation DUDDU UDUof system (b) with Theorems1 and8, we obtainφpDDUq FF FU DF DUFFFUD FU DF DUFUD. The proof is completed using the straightforward equation FUFUU FUF FUDon Motzkin paths.

Theorem 11. For n ¥ 0, the bijection φ from D^h,_n^¥ to M_n transports the statisticDDD as follows:

φpDDDq 2pUU UFq FDFUUFUF.

Proof. Combining the rst equation DD DDD UDD of system (b) with Theorems1and9, we obtainφpDDDq U UU UFFDUDFUUFUF. UsingUUF UD UU on Motzkin paths, the result holds.

Theorem 12. The bivariate generating functionsFppx, yq where the coecient of xⁿy^k is the number of Dyck paths in D_n^h,¥ containing exactly k occurrences of the pattern pP tU DU, U DD, DDU, DDDu are given by the following expres- sions listed in the same order as written above.

1 xpx²x²yyq ? A 2x²pxxy 1q , 1 xpx²yx²xy x1q ?

B 2x²pxyx 1q² , 1 xp2x²y²3x²y x² xyx1q ?

C 2x²ypxyx 1q , 1x x²y²x²yxy² x 1

? D 2x²pxyxyq²

where

A px 1q x²yx² xyx1

x³yx³2x²y 2x² xy 2x1 , B px 1q x²yx² 1

x³yx³3x²y 3x²3x 1 , C px 1q x²yx² 1

x³yx³3x²y 3x²3x 1 , D pxy 1q x²yx²xy x1

x³y²x³yx²y²2x²y 3x² 2xy x1 .

Proof. LetA_ppx, yq (resp. B_ppx, yq) be the bivariate generating function where the coecient ofxⁿy^kis the number of Motzkin paths of lengthnending with a at step (resp. down step) having exactlykoccurrences of the patterns related to the statistic φppq obtained in the r.h.s. of equations of Theorems8,9,10,11.

Using a renement of the classical decompositions of non-empty Motzkin paths by taking into account the occurrences of the considered patterns, we deduce functional equations forAppx, yq andBppx, yq. The method being classic, we do not give any more details. The solutions are obtained by a simple calculation.

ForpU DU:

$'

&

'%

F_ppx, yq 1 A_ppx, yq B_ppx, yq Appx, yq x xyAppx, yq xBppx, yq

B_ppx, yq x² x²yA_ppx, yq x²B_ppx, yq x²F_ppx, yqpF_ppx, yq 1q; ForpU DD:

$' ''

&

'' '%

Fppx, yq 1 Appx, yq Bppx, yq Appx, yq xFppx, yq

Bppx, yq x²yFppx, yq x²yAppx, yq x²Bppx, yq

x²Bppx, yq² x²yAppx, yqBppx, yq x²y²Appx, yq² x²yAppx, yqBppx, yq; ForpDDU:

$' ''

&

'' '%

F_ppx, yq 1 A_ppx, yq B_ppx, yq Appx, yq x xAppx, yq xyBppx, yq

B_ppx, yq x²F_ppx, yq x²yA_ppx, yqpF_ppx, yq 1q x²A_ppx, yq x²yBppx, yqFppx, yq;

ForpDDD:

$' ''

&

'' '%

Fppx, yq 1 Appx, yq Bppx, yq A_ppx, yq xF_ppx, yq

Bppx, yq x²Fppx, yq x²y²Bppx, yq x²y²Appx, yq{y x²y²Appx, yq²{y² x²y²A_ppx, yqB_ppx, yq{y x²y²B_ppx, yq² x²y²A_ppx, yqB_ppx, yq{y.

The above equations are intentionally left in non-simplied forms, in order to allow the reader to easily retrieve the rened decompositions from the classical decomposition of Motzkin pathsM MF MUMD.

Generating function G_ppxq for the popularity of a pattern p is obtained directly by evaluating ^BFp_B^p_y^x,yq

y1.Table 6provides the rst terms of the gen- erated sequences. See also Table 5 for an illustration of the distribution of pP tU DU, U DD, DDU, DDDu.

Unlike what happens for classical Dyck paths, the popularity of U DU in D^h,_n^¥₁ is equal to the popularity of U D in D^h,_n^¥, while the corresponding dis- tributions are dierent. Dyck paths from D_n^h,¥ avoiding a pattern U DU (resp.

DDD) are counted by the Generalized Catalan numbers (resp. by the num- bers of ordered trees withn edges and having no branches of length 1), which corresponds to the sequenceA004148(resp. A026418).

kzn 1 2 3 4 5 6 7 8 9 0 1 2 3 6 11 22 43 87 176 1 1 2 7 16 43 102 251 2 1 2 10 25 80 208

3 1 2 13 34 130

4 1 2 17 46

5 1 2 21

(a)DDD

kzn 1 2 3 4 5 6 7 8 9 0 1 1 2 4 8 17 37 82 185 1 1 1 3 7 17 41 102 252 2 1 1 4 10 28 73 200 3 1 1 5 13 41 113

4 1 1 6 16 56

5 1 1 7 19

(b)U DU kzn 1 2 3 4 5 6 7 8 9

0 1 2 3 4 5 6 7 8 9 1 1 5 14 31 59 102 164

2 2 14 57 174 444

3 4 39 209

4 9

(c)DDU

kzn 1 2 3 4 5 6 7 8 9 0 1 1 1 1 1 1 1 1 1 1 1 3 6 10 15 21 28 36 2 2 10 31 75 156 292

3 4 30 129 417

4 9 89

(d) U DD

Table 5: Number of paths from D_n^h,¥ having k occurences of the considered pattern.

Pattern Popularity sequence OEIS DDD 0,0,1,4,14,46,145,448,1365,4124,12387,37060

DDU 0,0,1,5,18,59,185,567,1715,5146,15363,45715 UDD 0,1,3,10,30,89,261,763,2227,6499,18973,55428

UDU 0,1,3,8,22,61,171,483,1373,3923,11257,32418 A025566 Table 6: Popularity of 3-length patterns inD^h,_n^¥ for1¤n¤12.

4 Acknowledgements

We would like to thank the anonymous referees for their helpful comments and suggestions.

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