1 Introduction and notations
The domain of lattice paths provides a very fertile ground for the combinatorial community. They have many applications in computer science, queuing theory, biology and physics , and there are a multitude of one-to-one correspondences with various combinatorial objects such as directed animals, pattern avoiding permutations, bargraphs, RNA structures and so on [4, 8, 17]. A recurring problem in combinatorics is the enumeration of these paths with respect to their length and other statistics [2, 3, 6, 11, 12, 13, 14, 16]. In the literature, Dyck and Motzkin paths are the most often considered, possibly because they are respectively counted by the famous Catalan and Motzkin numbers (see A108 and A1006 in the Sloane’s On-line Encyclopedia of Integer Sequences ).
Throughout this note, a lattice path is defined by a starting point , an ending point with , it consists of steps lying in , and it never goes below the -axis. The length of a path is the number of its steps. We denote by the empty path, i.e., the path of length zero. Constraining the steps to be in or , and fixing the end point on the -axis, we retrieve the well-known definition of Dyck and Motzkin paths  respectively. Let be the set of Dyck paths of semilength , we define . For short, we set , , and for .
Considering these notations, a Motzkin meander with catastrophes is a lattice path where possible steps are and for , such that all steps end on the -axis, and if we add the property that the path ends on the -axis, we call it a Motzkin excursion with catastrophes (see ). Dyck meanders and Dyck excursions with catastrophes are those avoiding the step . Let (resp. ) be the set of length Dyck meanders (resp. excursions) with catastrophes, and we set (resp. ). The sets of Motzkin meanders and excursions with catastrophes are respectively denoted by adding prime superscripts, and . As mentioned in Corollary 2.4 in , the cardinality of is given by the sequence A274115 in , and the cardinality of is given by the A224747. For instance, we have and , and we refer to Figure 1 for an illustration of these two paths. Since Motzkin meanders with catastrophes can be obtained from Dyck meanders with catastrophes by possibly adding flat steps , the ordinary generating function (o.g.f.) for the cardinality of is given by where is the o.g.f. for (see ), which generates the th term of A54391. Simarly, is counted by the th term of A54391.
Dyck meanders with catastrophes was first introduced by Krinik et al. in  in the context of queuing theory. They correspond to the evolution of the queue by allowing some resets modeled by a catastrophe step for . Recently in , Banderier and Wallner provide many results about the enumeration and limit laws of these objects. Using algrebraic methods they prove that the set of length Dyck meanders with catastrophes has the same cardinality as the set of equivalence classes of semilength Dyck paths modulo the positions of the pattern , which in turn (see ) is in one-to-one correspondence with the set of semilength Dyck paths avoiding occurrences at height of the patterns and . They also provide a constructive bijection between and the set of length Motzkin paths having their flat steps at height one.
The motivation of this work is to exhibit one-to-one correspondences between restricted Dyck paths (with no catastrophes) and the sets of paths with catastrophes , , , and . In Section 2 we present a constructive bijection between and . Considering its restriction to excursions with catastrophes, we prove that is in one-to-one correspondence with the set of Dyck paths in where any occurrence on the -axis appears before an occurrence of (not necessarily contiguous to the occurrence ). This bijection establishes a curious correspondence connecting Dyck meanders with catastrophes and equivalence classes modulo the positions of in Dyck paths. In Section 3 we conduct the counterpart study for Motzkin meanders and excursions. More precisely, we exhibit a bijection between and the set of semilength Dyck paths avoiding the pattern at height , which also induces a bijection from the set of Motzkin excursions with catastrophes to the set . The following table gives an overview of all these correspondences.
|Dyck meanders with cat.||Dyck paths avoiding and at|
|Dyck excursions with cat.||whose every on the -axis appears before|
|Dyck paths||Dyck paths starting with and avoiding and|
|Motzkin meanders with cat.||Dyck paths avoiding at|
|Motzkin excursions with cat.|
|Motzkin paths||Dyck paths avoiding at and at .|
2 Dyck meanders with catastrophes
In this section we exhibit a constructive bijection between the set of length Dyck meanders with catastrophes and the set of semilength Dyck paths having no occurrence of the consecutive three steps and at height (or equivalently with a minimal ordinate ). We set . Let us define recursively the map from to as follows. For , we set
where , and is either the empty path or a lattice path consisting of - and -steps such that (resp. , ) ends on the -axis in the case () (resp. (), ()), and does not necessarily end on the -axis in case ().
Due to the recursive definition, the image by of a length Dyck meander with catastrophes is a Dyck path of semilength . For instance, the images of , , , are respectively , , , . We refer to Figure 2 for an illustration of this mapping.
For any ,
- if then we have ,
- if then we have , where and for , the set consists of semilength Dyck paths avoiding the patterns and and starting with .
Proof. We proceed by induction on . The case is obvious. For , we assume that for any we have and for any we have . Now, let us prove the result for .
Whenever we can write where . Thus, we have , and using the recurrence hypothesis on and , is of semilength , starts with and avoids the pattern . Moreover (resp. ) is either empty or it starts with , which implies that avoids , and thus .
Now let us assume .
- If with , then and the recurrence hypothesis implies that avoids and at height .
- If where and , then the first part of the proof implies that avoids and , and with the recurrence hypothesis on , belongs to .
- If and , then using the first part of the proof is not empty and avoids and . The recurrence hypothesis implies that belongs to .
- If where and ends on the -axis, then using a simple induction on , is not empty and avoids and . The recurrence hypothesis implies that .
The induction is completed.
For , the map is a bijection. Moreover, we have .
Proof. Due to the enumerative results in  (see Corollary 2.4) and the above lemma, it suffices to prove that is injective from to . We proceed by induction on . The case is obvious. For , we assume that is an injection from to , and we prove the result for .
According to the definition of and Lemma 1, the image by of satisfying () is a Dyck path starting by for some where is a Dyck path in for some , which means that avoids ; a meander satisfying () is sent by to a Dyck path in for some ; a meander satisfying () is sent to a Dyck path starting with ; and a meander satisfying () is sent to a Dyck path starting with for some and such that it contains an occurrence on the -axis. Then, for , implies that and belong to the same case (), (), (), () or (). So, the recurrence hypothesis induces which completes the induction. Thus is injective. Since and have the same cardinality (see  and A274115 in ), is a bijection.
Considering the previous lemma, it suffices to check that is counted by the Catalan numbers in order to prove . A Dyck path is either empty or it consists of a sequence of where belongs to . Let be the generating function for the cardinality of (with respect to the semilength). We obtain the following functional equations which implies that is counted by the th Catalan number. Therefore is a bijection.
In , it is proven that the set is a representative set of the equivalence classes modulo the pattern on Dyck paths, i.e. two Dyck paths and are equivalent if and only if the positions of the occurrences are the same in and (see also ). So, the bijection establishes a direct correspondence between these classes and Dyck meanders with catastrophes.
Let be the subset of consisting of paths such that any occurrence on the -axis in appears before an occurrence of (not necessarily contiguous to the occurrence ). The next theorem gives a bijection between and the set of length Dyck excursions with catastrophes.
For , we have .
Proof. Thanks to Theorem 1, it suffices to check that for any , , and . Any satisfies one of the cases , , and with . We proceed by induction on the length in order to prove that . The case is obvious. Whenever satisfies the cases or , the only possibility for an occurrence of to appear at height zero in (resp. ) is to be inside . Applying the recurrence hypothesis on , . For a path satisfying the case , we have seen in the proof of Theorem 1 that starts necessarily with for followed by . Using the recurrence hypothesis for , we obtain . The induction is completed.
Now, let us prove that . Any path satisfies one of the following two cases: () does not contain any occurrence on the -axis, and () where and such that contains at least one occurrence of and avoids any occurrence on the -axis. Let (resp. ) be the set of Dyck paths in satisfying () (resp. ()), and let and be the corresponding generating functions for their cardinalities with respect to the semilength. Obviously, the generating function for satisfies
A nonempty path can be decomposed where and is a nonempty Dyck path avoiding and . Then either or with . Thus the generating function for is given by
Due to the form of a path , we deduce the functional equation
where is the generating function for and is the generating function for the paths in avoiding any occurrence on the -axis and containing at least one occurrence of . Due to Theorem 1, is also the o.g.f. for Dyck meanders with catastrophes that is . Then, we have where is the generating function for the set of Dyck paths in avoiding and on the -axis. Note that is exactly the set , then
Combining the previous equations, we obtain which is exactly the generating function of found by .
3 Motzkin meanders with catastrophes
In this section we exhibit a constructive bijection between the set of length Motzkin meanders with catastrophes and the set of semilength Dyck paths avoiding the patterns at height . Before defining this bijection we recall that there exists a one-to-one correspondence between length Motzkin paths and semilength Dyck paths avoiding . From a Dyck path avoiding , we replace each with , and we replace each remaining with . For instance, the image by of is (see [5, 7]).
Now, let us use in order to define recursively the map from to as follows. For , we set
where , , and are some possibly empty Motzkin paths (considering that is defined to be ).
Clearly, the image by of a length Motzkin meander with catastrophes is a Dyck path of semilength . For instance, the images by of , , , , are respectively , , , , . We refer to Figure 3 for an illustration of this mapping.
A simple observation provides the following results.
For , the map , defined above, induces a bijection from to . Moreover, the image of the set of length Motzkin paths is the set of semilength Dyck paths avoiding at height and the pattern at height one.
For , is the set Dyck paths in ending with , which implies that induces a one-to-one correspondence from paths to after deleting the last two steps from .
We would like to thank Cyril Banderier for suggesting us to explore constructive bijections between meanders with catastrophes and Dyck paths avoiding some patterns.
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