Counting pairs of noncrossing binary paths: A bijective approach

Discrete Mathematics 342 (2019) 352–359

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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

Counting pairs of noncrossing binary paths: A bijective approach ∗

K. Manes, I. Tasoulas, A. Sapounakis , P. Tsikouras Department of Informatics, University of Piraeus, 80, Karaoli & Dimitriou Str., 18534 Piraeus, Greece

article

info

a b s t r a c t

Article history: Received 29 May 2018 Received in revised form 10 October 2018 Accepted 10 October 2018 Available online xxxx

In this paper, a bijection between the set W of pairs of noncrossing binary paths and the set of ballot paths of odd length is constructed, mapping its subset which contains all pairs ending at the same point to the set of ballot paths ending at height 1. This bijection enables us to transfer statistics of W to natural statistics of ballot paths. © 2018 Elsevier B.V. All rights reserved.

Keywords: Pairs of noncrossing binary paths Ballot paths Dyck paths

1. Introduction Let Pn be the set of all (binary) paths P of length |P | = n, i.e., lattice paths P = p1 p2 · · · pn where each step pi , i ∈ [n], is either an upstep u = (1, 1) (also called rise) or a downstep d = (1, −1) (also called fall) and connects two consecutive points of the path P. A maximal sequence of u’s (resp. d’s) in P is called ascent (resp. descent) of P. The last point of an ascent (resp. descent) is called peak (resp. valley) of the path. Clearly, every peak (resp. valley) corresponds to either an occurrence of ud (resp. du), or an occurrence of u (resp. d) at the end of the path. It is convenient to consider that the starting point of a path is the origin of a pair ⋃ of axes. The y-coordinate of a lattice point on P is called height of this point. A peak at height 1 is called hill. We set P = n≥0 Pn , where P0 consists of only the empty path ε (the path which has no steps). A path that stays weakly above the x-axis (i.e., the heights of all its points are non-negative) is called ballot path. ⋃ We (k) denote by Bn (resp. Bn ) the set of all ballot paths of length n (resp. of length n ending at height k) and we set B = n≥0 Bn and B(k) =

(k)

⋃

n ≥k

Bn . In particular, ballot paths ending on the x-axis are called Dyck paths. Clearly, ballot paths are prefixes (0)

⋃

of Dyck paths. The set of Dyck paths of semilength n is denoted by Dn , i.e., Dn = B2n , and we set D = n≥0 Dn . (2n) 1 (see seq. A000108 in [13]). The corresponding generating The set Dn is counted by the Catalan number Cn = n+ 1 n function C (x) satisfies the equation C (x) = 1 + xC 2 (x), and the coefficients of its powers are given by the formula Cn,s = [xn ]C s (x) =

) , if s > 0, [n = 0], if s = 0,

{

2n+s s 2n+s n

(

where [S ] is the Iverson binary notation, i.e., for every proposition S, [S ] equals 1 if S is true and 0 if S is false. Clearly, since every ballot path P ∈ B(k) is decomposed uniquely as P = P0

k ∏

uPi ,

Pi ∈ D , 0 ≤ i ≤ k ,

i=1

∗ Corresponding author. E-mail address: [email protected] (A. Sapounakis). https://doi.org/10.1016/j.disc.2018.10.016 0012-365X/© 2018 Elsevier B.V. All rights reserved.

(1)

K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

353

(k)

k k+1 2 the generating function( of )the set Bn with respect to the length is equal ( nto )x C (x ). Then, using relation (1), it follows (k) k+1 n+1 easily that |Bn | = n+1 n−k , 0 ≤ k ≤ n and, by summation, that |Bn | = ⌊n/2⌋ (see seq. A053121 and seq. A001405 in [13]). 2

The set Dn can be partitioned with respect to many parameters, such as the number |P |τ of occurrences of a string τ in a path P (for details see [10,11]). In particular, for τ = ud it is well( known )( n ) that the number of Dyck paths of semilength n n having exactly k peaks is equal to the Narayana number Nn,k = 1k k k−1 (see seq. A001263 in [13]). The corresponding generating function N(x, y) satisfies the equation N(x, y) = 1 + xN 2 (x, y) + x(y − 1)N(x, y).

(2)

Moreover, it is also well known that if we partition Dn with respect to the number of return points (i.e., steps ending at height (2n−k) 0), then the number of Dyck paths of semilength n having k return points is equal to the Riordan number Rn,k = 2nk−k n (see seq. A033184 in [13]). It is easy to check (using identity 3.146 in [7]) that ⌊n/2⌋

∑

( Ri,k

)

n − 2i − 1

⌊n/2⌋ − i

i=k

( =

n−k−1

)

⌊n/2⌋ − k

.

(3)

Another interesting class of paths are the Motzkin paths, where we allow horizontal steps h = (1, 0) (also called flats), by the Motzkin numbers Mn (see seq. A001006 in [13]) that are given by the formula Mn = ( are )( ncounted ) ∑⌊n/2⌋which 1 n−j . The corresponding generating function M(x) satisfies the equation M(x) = 1 + xM(x) + x2 M 2 (x) and j=0 n−j j j+1 the coefficients of its powers are given by the formula n

{∑ ⌊n/2⌋

s

Mn,s = [x ]M (x) =

j=0

1,

n−j s n−j j

(

)(n+s−1) , j+s

if n, s > 0,

(4)

if n = 0, s ≥ 0.

A natural (partial) ordering on Pn is defined by the geometric representation of paths P , Q ∈ Pn , where P ≤ Q whenever P lies (weakly) below Q . It is easy to check that the poset (Pn , ≤) is a finite distributive lattice. A pair (P , Q ) with P ≤ Q is called pair of noncrossing paths. In the literature there exists a great number of papers concerning pairs or more generally k-tuples of noncrossing paths, (e.g., see [1,6,8,9,12]). (h) Let Wn be the set of all pairs (P , Q ) of noncrossing paths with |P | = |Q | = n, and let Wn be its subset consisting of the ⋃ pairs satisfying |Q |u − |P |u = h, i.e., the distance between their ending points is equal to 2h. We also set W = n≥0 Wn and (h)

W (h) = n≥0 Wn . L. Shapiro (see [12]) proved, using a recurrence formula, that

⋃

|Wn(h) | =

(

)

h + 1 2n + 2 n+1

n−h

,

(5)

which by summation gives that

|Wn | =

n ∑

|Wn(h) | =

h=0

(

2n + 1 n

) .

(6) (2h+1)

We note that the numbers given in relations (5) and (6) also count the sets B2n+1 and B2n+1 respectively (see seq. A039598 and seq. A001700 in [13]). Thus we have that, (2h+1) (0) |Wn | = |Bn+1 |, |Wn(h) | = |B2n +1 | and |Wn | = Cn+1 .

(7)

(0)

In the literature Wn , i.e., the set of all pairs of noncrossing paths of length n that end at the same point, has been studied (0) extensively and there exist bijections between Wn and Dn+1 (e.g., see [2,5]) although, as far as we know, an analogous result for Wn is not known. In Section 2, we give a bijective justification of equalities (7), by introducing a bijection θ from W to the set B(odd) of (0) (0) ballot paths of odd length, the restriction of which in Wn produces a new bijection between Wn and Dn+1 . Then, in Section 3, several enumeration results on W are established by transferring, via θ , certain statistics of W to natural statistics of ballot paths. 2. The bijection θ The definition of θ is recursive and it is based on associating a particular decomposition of W with a similar decomposition of B(odd). In order to describe these decompositions, we first define the notion of a common point and a joint step of two paths P = p1 p2 · · · pn and Q = q1 q2 · · · qn : Whenever the steps pi , qi , for some i ∈ [n], end at the same height, they have the same ending point, which is called a common point of P and Q . In addition, if pi = qi , then this step is called a joint step of P and Q . Note that, according to this definition, the initial point (0, 0) of the paths is not considered to be a common point. Every pair (P , Q ) ∈ W (0) \ {(ε, ε )} is decomposed uniquely with respect to the first common point as (uP ′ , uQ ′ )

or

(dP ′ , dQ ′ )

or

(dP ′ uP ′′ , uQ ′ dQ ′′ ),

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K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

Fig. 1. The bijection θ .

where (P ′ , Q ′ ), (P ′′ , Q ′′ ) ∈ W (0) . The first two cases occur whenever P and Q start with a joint step, so that their remaining parts P ′ and Q ′ clearly form a pair in W (0) . The third case occurs whenever the initial step is not a joint step, so that P must start with a d and Q with a u (since P ≤ Q ) and the paths must meet again with an upstep for P and a downstep for Q . If dP ′ u and uQ ′ d are their initial parts until their first common point, then dP ′ and uQ ′ have no common points and the distance between their ending points is 2, so that (P ′ , Q ′ ) ∈ W (0) . Obviously, the remaining parts P ′′ and Q ′′ also form a pair in W (0) . Furthermore, a pair (P , Q ) ∈ W \ W (0) is decomposed uniquely with respect to the last common point as (P ′ dP ′′′ , Q ′ uQ ′′′ ),

where (P ′ , Q ′ ) ∈ W (0) , (P ′′′ , Q ′′′ ) ∈ W .

Here, P ′ and Q ′ are the initial parts of P and Q until their last common point (these parts are empty iff no common point exists). The remaining parts dP ′′′ and uQ ′′′ have no common point so that (P ′′′ , Q ′′′ ) ∈ W . On the other hand, a path P ∈ B(1) \ {u} is decomposed uniquely with respect to the first step that returns at height 0 as udP ′

or uP ′ d or uP ′ ddP ′′ ,

where P ′ , P ′′ ∈ B(1) . In the first case, P returns at height 0 immediately after the first upstep. In the second case, P never returns at height 0. In the third case, P returns at height 0 after a nonempty sequence of steps above level 1 which is a Dyck path P ′ d. Furthermore, a path P ∈ B(odd) \ B(1) is decomposed uniquely with respect to the last upstep reaching height 2 as P ′ uP ′′′ ,

where P ′ ∈ B(1) , P ′′′ ∈ B(odd).

Indeed, since P does not end at height 1, this upstep reaching height 2 always exists, so that P ′ is a ballot path ending at height 1 and P ′′′ is a ballot path ending at odd height, since it starts at height 2, it never falls below it and P ends at odd height. We now define the mapping θ : W → B \ D as follows:

θ (ε, ε) = u, θ (uP ′ , uQ ′ ) = udθ (P ′ , Q ′ ), θ (dP ′ , dQ ′ ) = uθ (P ′ , Q ′ )d, θ (dP ′ uP ′′ , uQ ′ dQ ′′ ) = uθ (P ′ , Q ′ )ddθ (P ′′ , Q ′′ ), θ (P ′ dP ′′′ , Q ′ uQ ′′′ ) = θ (P ′ , Q ′ )uθ (P ′′′ , Q ′′′ ), where (P ′ , Q ′ ), (P ′′ , Q ′′ ) ∈ W (0) and (P ′′′ , Q ′′′ ) ∈ W , (see Fig. 1). We note that in the second, third and fifth cases of Fig. 1 the paths P ′ and Q ′ may have more than one common points. On the other hand, in the fourth case the paths dP ′ u and uQ ′ d have a unique common point (their ending point), whereas the paths P ′′ and Q ′′ may have more than one common points. Example. For the paths P = duduudduddud, Q = duuududuuddu we have that

θ (P , Q ) = θ (duduu, duuud) u θ (duddud, duuddu) = u θ (uduu, uuud) d u θ (duddu, duudd) u θ (ε, ε)

K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

355

Fig. 2. The image of the pair (P , Q ).

= u ud θ (duu, uud) d u u θ (uddu, uudd) d u u = u ud u θ (u, u) dd θ (ε, ε) d u u ud θ (ddu, udd) d u u = u ud u ud θ (ε, ε) dd u d u u ud u θ (d, d) dd θ (ε, ε) d u u = u ud u ud u dd u d u u ud u u θ (ε, ε) d dd u d u u = u ud u ud u dd u d u u ud u u u d dd u d u u, (see Fig. 2). Using induction on the length of the paths we can easily prove that θ is a bijection from W to B(odd), and that it satisfies the following properties, where we use the notation

|(P , Q )|(s,t) = #(s, t) in (P , Q ) = |{i ∈ [n] : pi = s and qi = t }|, for any s, t ∈ {u, d}. Properties of θ . For every (P , Q ) ∈ W we have that: P1 |θ (P , Q )| = 2|P | + 1. P2 |θ (P , Q )|u − |θ (P , Q )|d = 2(|Q |u − |P |u ) + 1. P3 If P , Q do not have a joint d, then # of joint u’s of P , Q = # of hills in θ (P , Q ), and # of common points of P , Q = # of returns in θ (P , Q ). Furthermore, for every (P , Q ) ∈ W (0) we have that P4 |P |u = |θ (P , Q )|du and |P |d = |θ (P , Q )|uu . P5 |(P , Q )|(u,u) = |θ (P , Q )|udu and |(P , Q )|(d,u) = |θ (P , Q )|ddu P6 P , Q have a joint d iff θ (P , Q ) ends with d. In this case, if (P , Q ) = (P1 dP2 , Q1 dQ2 ) is the first joint d decomposition of (P , Q ) (i.e., (P1 , Q1 ), (P2 , Q2 ) ∈ W (0) and P1 , Q1 do not have a joint d), then θ (P , Q ) = θ (P1 , Q1 )θ (P2 , Q2 )d. For completeness we give the proof of P6 (which is the most complicated one): Clearly the result holds when (P , Q ) = (ε, ε ). For a pair (P , Q ) ∈ W (0) \ {(ε, ε )} with a joint d, decomposed as (P , Q ) = (P1 dP2 , Q1 dQ2 ) according to the first joint d decomposition, we consider the following cases for (P1 , Q1 ): (ε, ε )

or

(uP1′ , uQ1′ )

or

(dP1′′ uP1′ , uQ1′′ dQ1′ )

where (P1′ , Q1′ ), (P1′′ , Q1′′ ) ∈ W (0) and P1′ , Q1′ have no joint d. Then using the induction hypothesis in each case, we have respectively that:

θ (P , Q ) = θ (ε dP2 , εdQ2 ) = θ (dP2 , dQ2 ) = uθ (P2 , Q2 )d = θ (ε, ε)θ (P2 , Q2 )d, θ (P , Q ) = θ (uP1′ dP2 , uQ1′ dQ2 ) = udθ (P1′ dP2 , Q1′ dQ2 ) = udθ (P1′ , Q1′ )θ (P2 , Q2 )d = θ (uP1′ , uQ1′ )θ (P2 , Q2 )d = θ (P1 , Q1 )θ (P2 , Q2 )d, θ (P , Q ) = θ (dP1′′ uP1′ dP2 , uQ1′′ dQ1′ dQ2 ) = uθ (P1′′ , Q1′′ )ddθ (P1′ dP2 , Q1′ dQ2 ) = uθ (P1′′ , Q1′′ )ddθ (P1′ , Q1′ )θ (P2 , Q2 )d = θ (dP1′′ uP1′ , uQ1′′ dQ1′ )θ (P2 , Q2 )d = θ (P1 , Q1 )θ (P2 , Q2 )d. Conversely we can similarly show that if P and Q have no joint d, then θ (P , Q ) does not end with d, thus completing the proof of P6. (k) (2k+1) (0) Clearly, Properties P1 and P2 justify that |Wn | = |B2n+1 | and |Wn | = |B2n+1 |. We note that the set Wn is mapped under θ to the set of ballot paths ending at the point (2n + 1, 1), which is exactly the set Dn+1 , after appending a downstep to each (0) ballot path, thus giving a new combinatorial justification of the formula |Wn | = Cn+1 .

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K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

3. Enumeration results In the following, we study several statistics of W , obtaining some new enumeration results. 3.1. Enumeration with respect to upsteps Proposition 3.1. The number of pairs (P , Q ) ∈ Wn with |P |u = k and |Q |u = ℓ, where k ≤ ℓ, is equal to

( )( ) ℓ−k+1 n+1 n+1 . n+1 l+1 k (0)

In particular, the number of pairs (P , Q ) ∈ Wn with |P |u = k is equal to the Narayana number Nn+1,k+1 . Proof. Let W (x, y, z) be the generating function of the set W and let F (x, y) be the generating function of the set W (0) , where x, y, z count the numbers |P |, |P |u , |Q |u respectively, for each pair of noncrossing paths (P , Q ). Clearly, since each (P , Q ) ∈ W \ W (0) is decomposed uniquely as (P , Q ) = (P ′ dP ′′′ , Q ′ uQ ′′′ ),

(P ′ , Q ′ ) ∈ W (0) , (P ′′′ , Q ′′′ ) ∈ W ,

we have that W (x, y, z) = F (x, yz) + xzF (x, yz)W (x, y, z), so that W (x, y, z) =

F (x, yz) 1 − xzF (x, yz)

.

By Property P4, it follows that the generating function F (x, y) also counts the number of Dyck paths of semilength n + 1 with k du’s, i.e., its coefficients are the Narayana numbers Nn+1,k+1 (see [3]). This gives that F (x, y) =

N(x, y) − 1 xy

,

where N(x, y) is the generating function of the Narayana numbers. Then, by Eq. (2) we obtain that F (x, y) = 1 + xF (x, y)(1 + y(1 + xF (x, y))). Then, a standard application of a version of the Lagrange inversion formula (see [3]) gives that

[xn yk ]F s =

s

(

)(

n+s

n+s k+s

)

n+s k

.

By expanding W into geometric series and after substituting the powers of F from the above equality, we easily deduce that

[xn yk z ℓ ]W (x, y, z) =

( )( ) ℓ−k+1 n+1 n+1 . □ n+1 l+1 k

As a special case of the above result, it follows that the set of all pairs of noncrossing Grand-Dyck paths of length 2n (i.e., paths ending at the point (2n, 0)) is equal to N2n+1,n+1 (see formula (2.2) in [1]). 3.2. Enumeration with respect to pairs of steps (h)

Proposition 3.2. The numbers an,k,h (resp. an,k ) of pairs in Wn (resp. Wn ) with k (u, u)’s are given by

( ) an,k,h =

n k

Mn−k−h,h+1 ,

an,k

n−k ⌋( )( ) ( ) ⌊∑ 2 n n−k n−k−j+1 , = k j j+1

j=0

where Mn,s are the powers( of ) the Motzkin numbers, given by relation (4). n In particular, an,k,0 = k Mn−k (Donaghey distribution, see seq. A091869 in [13]). Proof. We first show the above formulas for k = h = 0, i.e., for pairs (P , Q ) ∈ W (0) with no (u, u)’s. According to Property P5, the number of such pairs is equal to the number of Dyck paths of semilength n + 1 avoiding udu, which is known to be (h) equal to Mn (see [14]). Then, since any pair (P , Q ) ∈ Wn , h ≥ 1, is decomposed uniquely as (P , Q ) = (P1 dP2 · · · dPh+1 , Q1 uQ2 · · · uQh+1 ),

K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

357

where (Pi , Qi ) ∈ W (0) for all i ∈ [h + 1], it follows that an,0,h is the convolution of h + 1 copies of the Motzkin sequence. More precisely, an,0,h = [xn−h ]M h+1 (x) = Mn−h,h+1 . Furthermore, summing over all possible values of h, and after some manipulations, we obtain that an,0 =

n ∑

⌊n/2⌋ (

∑ n)(n − j + 1) . j j+1

Mn−h,h+1 =

h=0

j=0

(h)

(h)

For the general case, we observe that every element of Wn with k (u, u)’s is generated from a unique element (n) of Wn−k with no (u, u)’s, by inserting with repetition k (u , u)’s in the n − k + 1 possible positions. Since this is done in ways, we k (n) (n) obtain that an,k,h = k an−k,0,h and an,k = k an−k,0 , leading to the required formulas. □ Remarks. 1. If for a path P we denote by P its symmetrical path with respect to the vertical axis, then (P , Q ) ∈ W iff (Q , P) ∈ W and |(P , Q )|(d,d) = |(Q , P)|(u,u) . This shows that the numbers given in the previous proposition count also the number of pairs with k (d, d)’s. 2. The numbers an,k,h and an,k of Proposition 3.2 can be also obtained by the following equivalent formulas, using a different combinatorial approach: an,k,h =

an , k =

−k−h ⌋ ( ) ⌊ n∑ 2 n

k

h+1

(

2j + h

j+h+1

j=0

)(

j

n−k

)

2j + h

,

( )∑ )( ) n−k ( n j n−j . k ⌊j/2⌋ k j=0

(h)

For this, we first consider the subset An of Wn , the elements of which contain only (d, u)’s and (u, d)’s, and we set An = (h) An ∩ Wn . By replacing each (d, u) and (u, d) of (P , Q ) ∈ An by u and d respectively, we obtain a bijection between An and Bn , from which we get A(h) n

|

Bn(h)

|=|

(

|=

)

h+1 n+1

(8)

n−h 2

n+1

and

( |An | = |Bn | =

n

)

⌊n/2⌋

.

(9)

Furthermore, every element (P , Q ) ∈ Wn with k (u, u)’s can be obtained by a unique element of Aj for 0 ≤ j ≤ k, by inserting with repetition n − j pairs of steps in the j + 1 possible positions and then by selecting k of them to be (u, u)’s, (n)(n− ) j whereas the rest of them become (d, d)’s. Since this is done in j k ways, by formulas (8) and (9) we obtain an,k,h

( )( )( ) n−k ∑ h+1 j+1 n n−j = j−h j+1 j k 2 j=h j−h even

=

−k−h ⌋ ( ) ⌊ n∑ 2 n

k

j=0

h+1

(

2j + h

j+h+1

j

)(

n−k

)

2j + h

and an , k

)( )( ) ( )∑ )( ) n−k ( n−k ( ∑ j n n−j n j n−j = = . j k k ⌊j/2⌋ k ⌊j/2⌋ j=0

j=0

The same arguments as in the previous remark can be used for the proof of the next result. Proposition 3.3. The number of pairs in Wn having ℓ (d, u)’s and k (u, d)’s, where k ≤ ℓ, is equal to

( )( ) ℓ − k + 1 n n − k n−k−ℓ 2 . ℓ+1 k ℓ (0)

In particular, the number of pairs in Wn having k (d, u)’s is equal to

(n) 2k

Ck 2n−2k (Touchard distribution, seq. A091894 in [13]). (0)

We note that by the second equality of Property P5, we can immediately deduce that the number of (P , Q ) ∈ Wn having k (d, u)’s is equal to the number of Dyck paths of semilength n with k ddu’s, which is known to be equal to the Touchard number (see [3]).

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K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

3.3. Enumeration with respect to joint steps and common points In this section we study the parameters ‘‘number of common points’’ (where the starting point of the paths is not taken into account) and ‘‘joint steps’’. For this, we use some more or less known results on Dyck and ballot paths. It is well known that using the first return decomposition (2n−k) of Dyck paths it follows easily that the number of P ∈ Dn with k returns is equal to the Riordan number Rn,k = 2nk−k n (see [3,4]). Similarly, it can be proved that the number of P ∈ Dn with k returns and no hills is equal to Rn,2k . Furthermore, since every ballot path P ∈ B \ D is decomposed uniquely in the form P = P1 uP2 , where P1 ∈ D, P2 ∈ B, it follows that P , P1 have the same returns so that, by concatenation and ( n−using ) k−1 identity (3), it can be easily shown that the number of P ∈ Bn with k returns (resp. k returns and no hills) is equal to ⌊n/2⌋−k

( n−2k−1 )

(see seq. A191528 in [13]) (resp. ⌊n/2⌋−2k ). On the set B(odd) we consider the parameter ld1 : ‘‘length of the descent reaching the last (rightmost) point at height 1’’, which takes zero value iff this point is reached by an upstep. It is well known that the parameters nr: ‘‘number of returns’’ and ld: ‘‘length of the last descent’’ are equidistributed on D (e.g., see [4]). Similarly, the parameters nr and ld1 are equidistributed on B(odd). To see this, consider ψ : B(odd) → B(odd) defined by

ψ (u) = u, ψ (P1 du) = uψ (P1 )d, ψ (uP2 d) = ψ (P2 )du, ψ (P1 duP2 d) = ψ (P2 )duψ (P1 )d, ψ (P1 uP3 ) = ψ (P1 )uP3 , where P1 , P2 ∈ B(1) and P3 ∈ B(odd). We can easily check by induction that ψ is an involution that preserves the length, maps the set B(2h+1) to itself and sends the one of the above two parameters to the other. Using this equidistribution and the enumeration results with respect to the number of returns (already mentioned in the beginning of this section), we obtain the following result. (1)

Lemma 3.4. The number of ballot paths P ∈ B2n+1 (resp. P ∈ B2n+1 ) with ld1 (P) = k is equal to Rn+1,k+1 (resp.

(2n−k) n

).

(0)

Proposition 3.5. The number of pairs in Wn (resp. Wn ) with: (i) j joint d’s is equal to

( Rn+1,j+1 (resp.

2n − j n

) , see also seq. A092392 in [13])

(ii) k common points and j joint d’s is equal to

(k)

Rn−j,k−j (resp.

j

),

(k)(2n−j−k−1) j

n−k

(iii) k common points, i joint u’s and j joint d’s is equal to

( )(

)

k

k−j

j

i

( )( Rn−i−j,2(k−i−j) (resp.

k

k−j

j

i

)(

2n − 2k − 1

)

n − 2k − i − j

).

Proof. (i) According to the previous lemma, it is enough to show that P , Q have j joint d’s iff ld1 (θ (P , Q )) = j, for every (P , Q ) ∈ W . For this, we use Property P6 of θ . Firstly, if (P , Q ) ∈ W (0) , then, if j = 0, θ (P , Q ) ends with u so that ld1 (θ (P , Q )) = 0, whereas if j > 0 then (P , Q ) is decomposed uniquely as (P , Q ) = (P1 dP2 d · · · Pj dPj+1 , Q1 dQ2 d · · · Qj dQj+1 ), where (Pi , Qi ) ∈ W (0) and Pi , Qi have no joint d’s for all i ∈ [j + 1], so that

θ (P , Q ) = θ (P1 , Q1 )θ (P2 , Q2 ) · · · θ (Pj+1 , Qj+1 )dj . Since each θ (Pi , Qi ) ends at height 1, we deduce that ld1 (θ (P , Q )) = j. Finally, if (P , Q ) ∈ W \ W (0) , then, using the (last common point) decomposition (P , Q ) = (P1 dP2 , Q1 uQ2 ), where (P1 , Q1 ) ∈ W (0) , (P2 , Q2 ) ∈ W and P1 , Q1 have j joint d’s, we have that ld1 (θ (P , Q )) = ld1 (θ (P1 , Q1 )uθ (P2 , Q2 )) = ld1 (θ (P1 , Q1 )) = j. (ii) We first show the result for j = 0. If (P , Q ) ∈ W and P , Q have no joint d’s, then, by Property P3, θ maps the k common points to the k returns of θ (P , Q ) and, according to (i), ld1 (θ (P , Q )) = 0, i.e., the last point at height 1 is reached by an upstep. (0) By deleting this upstep, we obtain an arbitrary path in Dn (resp. B2n ) whenever (P , Q ) ∈ Wn (resp. (P , Q ) ∈ Wn ), while preserving the number of returns. (2n−Then, ) the result follows, since the number of paths in Dn (resp. B2n ) having k returns is k−1 known to be equal to Rn,k (resp. n−k ). For the case j > 0, it is enough to see that paths in Wn with j joint points and k common points are constructed from paths in Wn−j with no joint d’s and k − j common points, by inserting with repetition j joint d’s in k − j + 1 possible positions. (iii) The proof is similar to the proof of (ii) and it is omitted. We note that the only difference is that in this case the result is shown firstly for i = j = 0, where the ballot path θ (P , Q ) is in this case hill-free. □

K. Manes et al. / Discrete Mathematics 342 (2019) 352–359

359

Remark. The items (i) (resp. (ii)) of the previous proposition can be also obtained algebraically by summing the results of items (ii) (resp. (iii)) in terms of k (resp. i). (0) By summing in terms of k in (iii) of Proposition 3.5, we obtain that the number of pairs in Wn (resp. Wn ) with i joint u’s and j joint d’s is equal to bn,j =

∑⌊ n−2 j ⌋ (r +j)(2(n−j−r)−1) r =0

j

n−j−2r

(i+j)

(i+j)

j

j

an,i+j (resp.

bn,i+j ), where an,j =

∑⌊ n−2 j ⌋ (r +j) r =0

j

Rn−j,2r (see seq. A065600 in [13]) and

(see seq. A158815 in [13]). (0)

It follows easily that the number of pairs in Wn (resp. Wn ) with j joint steps is equal to 2j an,j (resp. 2j bn,j ). We close by giving a new bijection, using θ , between W and the set G (d) of Grand-Dyck paths starting with d. 3.4. A mapping from noncrossing paths to Grand-Dyck paths Since, by adding an extra downstep at the beginning of a ballot path ending at height 1, we get a Grand-Dyck path, it is natural to define φ (P , Q ) = dθ (P , Q ), for all (P , Q ) ∈ W (0) . This is a bijection between W (0) and the subset of G (d) which contains all paths with depth 1. (The depth of a path is the absolute value of the height of its lowest point.) In order to extend φ on W , we first notice that every P ∈ G (d) is decomposed uniquely as P = dRB, where R ∈ G (d), B ∈ B(1) . Next, we extend φ on W by defining

φ (P ′ dP ′′ , Q ′ uQ ′′ ) = dφ (P ′′ , Q ′′ )θ (P ′ , Q ′ ), where (P ′ , Q ′ ) ∈ W (0) , (P ′′ , Q ′′ ) ∈ W . It is easy to check that φ is a bijection between W and G (d) such that |φ (P , Q )| = 2(|P | + 1) and the depth of φ (P , Q ) equals |Q |u − |P |u + 1. From the above we obtain the following result. 3.6. (Proposition ) ( The ) number of Grand-Dyck paths of semilength n (resp. semilength n and depth k) starting with d is equal to 2n−1 k 2n n

(resp.

n n−k

).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

W.Y.C. Chen, S.X.M. Pang, E.X.Y. Qu, R.P. Stanley, Pairs of noncrossing free dyck paths and noncrossing partitions, Discrete Math. 309 (2009) 2834–2838. M.-P. Delest, G. Viennot, Algebraic languages and polyominoes enumeration, Theoret. Comput. Sci. 34 (1984) 169–206. E. Deutsch, Dyck path enumeration, Discrete Math. 204 (1999) 167–202. E. Deutsch, An involution on Dyck paths and its consequences, Discrete Math. 204 (1999) 163–166. E. Deutsch, L. Shapiro, A survey of the fine numbers, Discrete Math. 241 (2001) 241–265. S. Elizalde, Bijections for pairs of non-crossing lattice paths and walks in the plane, European J. Combin. 49 (2015) 25–41. H. Gould, Combinatorial Identities, Morgantown, W. Va, 1972. A. Guttmann, A. Owczarek, X. Viennot, Vicious walkers and Young tablaeux I: without walls, J. Phys. A 31 (1998) 8123–8135. C. Krattenthaler, A. Guttmann, X. Viennot, Vicious walkers, friendly walkers and Young tableaux II: with a wall, J. Phys. A 33 (2000) 8835–8866. K. Manes, A. Sapounakis, I. Tasoulas, P. Tsikouras, General results on the enumeration of strings in dyck paths, Electron. J Combin. 18 (2011) #P74. A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in dyck paths, Discrete Math. 307 (2007) 2909–2924. L. Shapiro, A catalan triangle, Discrete Math. 14 (1976) 83–90. N.J.A. Sloane, et al., The on-line encyclopedia of integer sequences. http://oeis.org. Y. Sun, The statistic number of udu’s in Dyck paths, Discrete Math. 287 (2004) 177–186.