ISRN Combinatorics

Volume 2013 (2013), Article ID 107454, 6 pages

http://dx.doi.org/10.1155/2013/107454

Research Article

Permutations and Pairs of Dyck Paths

Marilena Barnabei,¹ Flavio Bonetti,¹ and Matteo Silimbani²

¹Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

²LaBRI, Université de Bordeaux, 351 Cours de la Libération, 33405 Talence, France

Received 7 November 2012; Accepted 26 November 2012

Academic Editors: E. Manstavicius, S. Natanzon, and Y. Shi

Copyright © 2013 Marilena Barnabei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We define a map between the symmetric group and the set of pairs of Dyck paths of semilength . We show that the map is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.

1. Introduction

We say that a permutation contains a pattern if contains a subsequence that is order-isomorphic to . Otherwise, we say that avoids . Given a pattern , denote by the set of permutations in avoiding .

The sets of permutations that avoid a single pattern have been completely determined in last decades. More precisely, it has been shown [1] that, for every , the cardinality of the set equals the th Catalan number, which is also the number of Dyck paths of semilength (see [2] for an exhaustive survey). Many bijections between , , and the set of Dyck paths of semilength have been described (see [3] for a fully detailed overview).

The case of patterns of length appears much more complicated, due both to the fact that the patterns are not equidistributed on , and the difficulty of describing bijections between , , and some set of combinatorial objects.

In this paper we study the case . An explicit formula for the cardinality of has been computed by I. Gessel (see [2, 4]).

We present a bijection between and a set of pairs of Dyck paths of semilength . More specifically, we define a map from to the set of pairs of Dyck paths, prove that every element in the image of corresponds to a single element in , and characterize the set of all pairs that belong to the image of the map .

2. Dyck Paths

A Dyck path of semilength is a lattice path starting at , ending at , and never going below the -axis, consisting of up steps and down steps . A return of a Dyck path is a down step ending on the -axis. A Dyck path is irreducible if it has only one return. An irreducible component of a Dyck path is a maximal irreducible Dyck subpath of .

A Dyck path is specified by the lengths of its ascents (viz., maximal sequences of consecutive up steps) and by the lengths of its descents (maximal sequences of consecutive down steps), read from left to right. Set and . If is the semilength of , we have of course , hence the Dyck path is uniquely determined by the two sequences and . The pair is called the ascent-descent code of the Dyck path .

Obviously, a pair , where and , is the ascent-descent code of some Dyck path of semilength if and only if

(i); (ii); (iii); (iv) for every .

It is easy to check that the returns of a Dyck path are in one-to-one correspondence with the indices such that . Hence, a Dyck path is irreducible whenever we have for every .

For example, the ascent-descent code of the Dyck path in Figure 1 is , where and . Note that and . In fact, is irreducible.

Figure 1

We describe an involution due to Kreweras (a description of this bijection, originally defined in [5], can be found in [6]) and discussed by Lalanne (see [7, 8]) on the set of Dyck paths. Given a Dyck path , the path can be constructed as follows:

(i)if is the empty path , then ; (ii)if is nonempty:

(a)flip the Dyck path around the -axis, obtaining a path ; (b)draw northwest (resp. northeast) lines starting from the midpoint of each double descent (resp. ascent); (c)mark the intersection between the th northwest and th northeast line, for every ; (d) is the unique Dyck path that has valleys at the marked points (see Figure 2).

We define a further involution on the set of Dyck paths, which is a variation of the involution , as follows:

(i)if is the empty path , then ;

(a)consider a Dyck path and flip it with respect to a vertical line; (b)decompose the obtained path into its irreducible components ; (c)replace every component with in order to get (see Figure 3).

We point out that the map appears in a slightly modified version in the paper [6].

Figure 2: The map .

Figure 3: The map .

We now give a description of the map in terms of ascent-descent code. Obviously, it is sufficient to consider the case of an irreducible Dyck path .

Let be the ascent-descent code of an irreducible path of semilength , with and . Straightforward arguments show that the ascent-descent code of can be described as follows:

(i)set and set , where the ’s are written in decreasing order. Then, ; (ii)consider the set , where the ’s are written in decreasing order. Then, .

Finally, we introduce an order relation on the set of Dyck paths of the same semilength. This order relation will be defined in three steps:

(i) consider two irreducible Dyck paths and of semilength . Let be the ascent-descent code of , with and . We say that covers in the relation if the ascent code of is obtained by removing an integer from and the descent code of is obtained by removing an integer from , with .  Roughly speaking, covers if it can be obtained from by “closing” the rectangles corresponding to an arbitrary collection of consecutive valleys of (see Figure 4); (ii) the desired order relation on the set of irreducible Dyck paths is the transitive closure of the above covering relation; (iii) the relation is extended to the set of all Dyck path of a given semilength as follows: if and are two arbitrary Dyck paths and and are their respective decompositions into irreducible parts, then whenever and for every .

We point out that the described order relation is a subset of the inclusion order relation defined in [9]. In the following sections, we will show that the defined relation is more suited for our studies.

Figure 4: The dotted Dyck path covers the solid one.

3. LTR Minima and RTL Maxima of a Permutation

Some of the well-known bijections between , , and the set of Dyck paths of semilength (see [10–12]), are based on the two notions of left-to-right minimum and right-to-left maximum of a permutation :

(i)the value is a left-to-right minimum (LTR minimum for short) at position if for every ; (ii)the value is a right-to-left maximum (RTL maximum) at position if for every .

For example, the permutation

has the LTR minima 5, 3, 2, and 1 (at positions , , , and ) and RTL maxima and (at positions and ).

We denote by and the sets of values and positions of the LTR minima of , respectively. Analogously, and denote the sets of values and positions of the RTL maxima of .

Recall that the reverse-complement of a permutation is the permutation defined by

For example, consider the permutation

Then

Note that the sets and are closed under reverse-complement, namely, (resp., ) if and only if (resp. ).

The first assertion in the next theorem goes back to the seminal paper [12], while the second one is an immediate consequence of the straightforward fact that is a LTR minimum of a permutation at position if and only if   is RTL maximum of at position .

Theorem 1. A permutation is completely determined by the two sets and of values and positions of its left-to-right minima. A permutation in is completely determined, as well, by the two sets and of values and positions of its right-to-left maxima.

Also -avoiding permutations can be characterized in terms of LTR minima and RTL maxima.

This characterization can be found in [2] and is based on an equivalence relation on defined as follows: and share the values and the positions of LTR minima and RTL maxima.

For example,

Straightforward arguments lead to the following result stated in [2].

Theorem 2. Every equivalence class of the relation contains exactly one -avoiding permutation. In this permutation, the values that are neither LTR minima nor RTL maxima appear in decreasing order.

4. The Maps and

We define two maps and between and the set of Dyck paths of semilength . Given a permutation , the path is contructed as follows:

(i)decompose as , where are the left-to-right minima in and are (possibly empty) words; (ii)set ; (iii)read the permutation from left to right and translate any LTR minimum () into up steps and any subword into down steps, where denotes the number of elements in .

The statement of Theorem 1 implies that the map is a bijection when restricted to .

Note that the ascent-descent code of the path is obtained as follows:

(i); (ii), where is the position of .

We define a further map :

(i)decompose as , where are the right-to-left maxima in and are (possibly empty) words;(ii)set ; (iii)associate with () the steps ;(iv)associate with each entry in a step.

Also in this case, the map is a bijection when restricted to .

The ascent-descent code of the path is obtained as follows:

(i); (ii), where is the position of .

In Figure 5 the two paths and corresponding to are shown.

Figure 5: The Dyck paths corresponding to .

We can now define a map , setting

The statement of Theorem 2 implies that the map is injective when restricted to .

Note that the map behaves properly with respect to the reverse-complement and the inversion operators.

Proposition 3. Let be a permutation in . One has:

(i), hence, the permutation is -invariant if and only if . (ii)( ), where is the path obtained by flipping with respect to a vertical line. Hence, the permutation is an involution if and only if both and are symmetric with respect to a vertical line.

For example, consider . The two paths associated with are shown in Figure 5. The permutation is associated with the two paths in Figure 6, while the permutation corresponds to the two paths in Figure 7.

Figure 6: The Dyck paths corresponding to .

Figure 7: The Dyck paths corresponding to .

Moreover, the map has the following further property that will be crucial in the proof of our main result.

Recall that a permutation is said to be right-connected if it does not have a suffix of length , that is a permutation of the symbols .

For example, the permutation

is right-connected, while

is not right-connected.

According to this definition, we can split every permutation into right-connected components:

Note that, if a permutation is not right-connected, is the juxtaposition of a permutation of the set and the permutation of the set .

Proposition 4. Let be a non right-connected permutation in , with , where is a permutation of the set and is a permutation of set of the set . Then

with and , .

The order relation on Dyck paths defined in Section 2 can be exploited to define two order relations on the set as follows:

(i) if and only if ; (ii) if and only if .

These order relations can be intrinsically described as follows.

Proposition 5. Let . One has whenever:

(i) ; (ii) ; (iii) setting:  (written in decreasing order),   (in decreasing order),   (in increasing order),   then for every .

Similarly, whenever:

(i) ; (ii) ; (iii) setting:   (written in increasing order),  (in increasing order),   (in decreasing order),  then for every .

For example, consider the permutation

We have , , , and . The permutation

is such that and , hence, . Moreover, the permutation

is such that and , hence, .

5. Main Results

We say that a pair of Dyck paths is admissible if there exists a permutation such that and . Needless to say, the set of admissible pairs is in bijection with the set of -avoiding permutations.

In the case when the two paths and are irreducible, if the pair is admissible, then the peaks of the two paths have different and coordinates. We observe that this is not a sufficient condition. For example, consider the pair and . The unique permutation having LTR-minima and RTL-maxima at the positions prescribed by and has an extra LTR-minimum at position . Hence, is not admissible.

We want to show that the operator on Dyck paths allows us to characterize the set of admissible pairs. We begin with a preliminary result concerning the pairs of Dyck paths corresponding to -avoiding permutations:

Theorem 6. For every , one has:

Proof. Proposition 4, together with the definition of the map , allows us to restrict our attention to the right-connected case.

Recall (see [12]) that a permutation avoids if and only if the set . It is simple to check that, if is right-connected, the sets of LTR minima and RTL maxima are disjoint.

Consider now a permutation with LTR minima and RTL maxima . Denote by the ascent-descent code of the path and by the ascent-descent code of the path .

As noted before, the ascent code of is obtained by computing the integers and then considering the set , which can be written as

Since , we have

Hence, .

Similarly, the descent code of is obtained by considering the set

Since , we have

Hence, .

For example, the -avoiding permutation corresponds to the pair of Dyck paths in Figure 3.

We are now in position to state our main result.

Theorem 7. A pair is admissible if and only if and .

Proof. Consider a permutation and let be the unique permutation in with the same LTR minima as , at the same positions. Obviously, , since in every element that is not a LTR minimum is a RTL maximum (see Proposition 5). Recalling that , we get the first inequality. The other inequality follows from the fact that the pair is admissible whenever the pair is admissible.

Consider now a pair of Dyck paths such that and . Proposition 4 allows us to restrict to the case irreducible. Denote by and the permutations in corresponding via to the pairs and , respectively. Since and , we have and .

We define a permutation as follows:

(i) if ; (ii) if ; (iii)if , we have , written in increasing order. Set

where are the elements in , written in decreasing order.

The permutation is obtained as the concatenation of three decreasing sequences. Hence, avoids . We have to prove that and .

It is immediate that . In order to prove that it remains to show that the values are not LTR minima of .

In fact, for every , consider . Consider the sets , , and their subsets and . The elements in do not belong to (and hence, the elements in are the largest elements in ). Proposition 5 ensures that each of them occupies in a position that is strictly greater than the position occupied in . This implies that and that at most elements in occupy in a position that belongs to . Hence, in , at least positions in are occupied by entries belonging to . This implies that there is in a position preceding occupied by a value less than . Hence, is not a LTR minimum of .

Analogous arguments can be used to prove that . Hence, , as desired.

For example, consider the pair of Dyck paths in Figure 8.

Figure 8

It can be checked that and . The permutations and are as follows:

We have , , , , , , , and .

The permutation is

As expected, , , , and .

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