Permutations and Pairs of Dyck Paths

We say that a permutation σσ σ SSnn contains a pattern ττ σ SSkk if σσ contains a subsequence that is order-isomorphic to ττ. Otherwise, we say that σσ avoids ττ. Given a pattern ττ, denote by SSnn(τττ the set of permutations in SSnn avoiding ττ. e sets of permutations that avoid a single pattern ττ σ SS3 have been completely determined in last decades. More precisely, it has been shown [1] that, for every ττ σ SS3, the cardinality of the set SSnn(τττ equals the nnth Catalan number, which is also the number of Dyck paths of semilength nn (see [2] for an exhaustive survey). Many bijections between SSnn(τττ, ττ σ SS3, and the set of Dyck paths of semilength nn have been described (see [3] for a fully detailed overview). e case of patterns of length 4 appears much more complicated, due both to the fact that the patterns ττ σ SS4 are not equidistributed on SSnn, and the difficulty of describing bijections between SSnn(τττ, ττ σ SS4, and some set of combinatorial objects. In this paper we study the case ττ τ ττ34. An explicit formula for the cardinality of SSnn(ττ34τ has been computed by I. Gessel (see [2, 4]). We present a bijection between SSnn(ττ34τ and a set of pairs of Dyck paths of semilength nn. More speci�cally, we de�ne a map νν from SSnn to the set of pairs of Dyck paths, prove that every element in the image of νν corresponds to a single element in SSnn(ττ34τ, and characterize the set of all pairs that belong to the image of the map νν. 2. Dyck Paths


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 . e sets of permutations that avoid a single pattern    3 have been completely determined in last decades.More precisely, it has been shown [1] that, for every    3 , 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   (,    3 , and the set of Dyck paths of semilength  have been described (see [3] for a fully detailed overview).
e case of patterns of length 4 appears much more complicated, due both to the fact that the patterns    4 are not equidistributed on   , and the difficulty of describing bijections between   (,    4 , and some set of combinatorial objects. In this paper we study the case   34.An explicit formula for the cardinality of   (34 has been computed by I. Gessel (see [2,4]).
We present a bijection between   (34 and a set of pairs of Dyck paths of semilength .More speci�cally, we de�ne 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   (34, and characterize the set of all pairs that belong to the image of the map .

Dyck Paths
A Dyck path of semilength  is a lattice path starting at (0, 0, ending at (, 0, 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 speci�ed 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 le 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     , … ,   .e 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) 0 <   ; (ii)     <   < ⋯ <      ; (iii)     <   < ⋯ <      ; (iv)   ≥   for every       .It is easy to check that the returns of a Dyck path are in oneto-one correspondence with the indices 1 ≤  ≤  such that   =   .Hence, a Dyck path is irreducible whenever we have   >   for every 1 ≤  ≤   1.
For example, the ascent-descent code of the Dyck path  in Figure 1 is ( , where  =   and  =  .Note that  1 >  1 and   >   .In fact,  is irreducible.
We describe an involution  due to Kreweras (a description of this bijection, originally de�ned 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) �ip 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 de�ne 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 �ip 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 modi�ed version in the paper [6].
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  =  1  …   ℎ and  =  1  …   ℎ .Straightforward arguments show that the ascentdescent code ( ′   ′  of  ′ ( can be described as follows: (i) set   =   𝑖  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  =  1   ⋯   and  =  1   ⋯   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 de�ned in [9].In the following sections, we will show that the de�ned relation is more suited for our studies.

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][11][12]), are based on the two notions of le-to-right minimum and right-to-le maximum of a permutation  =  1   ⋯   : (i) the value   is a le-to-right minimum (LTR minimum for short) at position  if   <   for every  < ; (ii) the value   is a right-to-le maximum (RTL maximum) at position  if   >   for every  > .
We denote by  min ( and  min ( the sets of values and positions of the LTR minima of , respectively.Analogously,  max ( and  max ( denote the sets of values and positions of the RTL maxima of .
Recall that the reverse-complement of a permutation     is the permutation de�ned by For example, consider the permutation   2 4 7 3 1 8 9 5 6. en Note that the sets   (123 and   (1234 are closed under reverse-complement, namely,     (123 (resp.,     (1234) if and only if  rc    (123 (resp. rc    (1234).e �rst 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   1   is RTL maximum of  rc at position   1  .eorem 1.A permutation     (123 is completely determined by the two sets  min ( and  min ( of values and positions of its le-to-right minima.A permutation in   (123 is completely determined, as well, by the two sets  max ( and  max ( of values and positions of its right-to-le maxima.
Also 1234-avoiding permutations can be characterized in terms of LTR minima and RTL maxima.
is characterization can be found in [2] and is based on an equivalence relation on   de�ned 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].(ii) set  0  0; (iii) associate with   ( > 0) the steps     1 ; (iv) associate with each entry in   a  step.
In Figure 5  e statement of eorem 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) ()  ( )  ( rc )  ( ), hence, the permutation  is rc-invariant if and only if   .
(ii) ()  ( )  ( − )  (rev() rev()), where rev() is the path obtained by �ipping  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         .e two paths associated with  are shown in Figure 5. e permutation  rc         is associated with the two paths in Figure 6, while the permutation  −         corresponds to the two paths in Figure 7.
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 rightconnected if it does not have a suffix  ′ of length   , that is a permutation of the symbols   …  .
For example, the permutation

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 1234-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   .e unique permutation   3 2 1 5 4 having LTR-minima and RTL-maxima at the positions prescribed by  and  has an extra LTR-minimum at position 2. 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 123-avoiding permutations: eorem 6.For every     (123, one has: Proof.Proposition 4, together with the de�nition of the map  ′ , allows us to restrict our attention to the right-connected case.
Recall (see [12]) that a permutation  avoids 123 if and only if the set  min (   max (  .It is simple to check that, if  is right-connected, the sets of LTR minima and RTL maxima are disjoint.
As noted before, the ascent code  ′ of  ′ ( is obtained by computing the integers      𝑘 Hence,  ′   * .
We are now in position to state our main result.
. The Maps  and  We de�ne two maps  and  between   and the set   of Dyck paths of semilength .Given a permutation     , the path ( is contructed as follows: (i) decompose  as    1  1  2  2 ⋯     , where  1 ,  2 , … ,   are the le-to-right minima in  and  1 ,  2 , … ,   are (possibly empty) words; (ii) set  0    1; (iii) read the permutation from le to right and translate any LTR minimum   ( > 0) into  1    up steps and any subword   into    1 down steps, where   denotes the number of elements in   .    1   1 ,   1   2 , … ,   1   1 ; (ii)    2  1,  3  1, … ,    1, where   is the position of   .We de�ne a further map     →   : (i) decompose  as    ℎ  ℎ  ℎ1  ℎ1 ⋯  1  1 , where  1 ,  2 , … ,  ℎ are the right-to-le maxima in  and  1 ,  2 , … ,   are (possibly empty) words; 2. Every equivalence class of the relation  contains exactly one 1234-avoiding permutation.In this permutation, the values that are neither LTR minima nor RTL maxima appear in decreasing order.4estatement of eorem 1 implies that the map  is a bijection when restricted to   (123.Note that the ascent-descent code (,  of the path ( is obtained as follows:(i) Note that, if a permutation  is not right-connected,  is the juxtaposition of a permutation ′′of the set {   …   and the permutation  ′ of the set { …  .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 { …  .en  ()       ()       (10) with    (  ) and    (  ),    .   if and only if ()  ().ese order relations can be intrinsically described as follows.Let      .One has     whenever:  min ()  {   …   ℎ  (written in decreasing order),  min ()   min ()  {        …      (in decreasing order),  max ()  {   …     (written in increasing order),  max (   max (    1 ,   2 , … ,    } (in increasing order),  max (   max (    1 ,   2 , … ,    } (in decreasing order), then   <   for every . max (  9} and  max (  9}, hence,   .