Labelled well-quasi-order in juxtapositions of permutation classes

The juxtaposition of permutation classes $\mathcal{C}$ and $\mathcal{D}$ is the class of all permutations formed by concatenations $\sigma\tau$, such that $\sigma$ is order isomorphic to a permutation in $\mathcal{C}$, and $\tau$ to a permutation in $\mathcal{D}$. We give simple necessary and sufficient conditions on the classes $\mathcal{C}$ and $\mathcal{D}$ for their juxtaposition to be labelled well-quasi-ordered (lwqo): namely that both $\C$ and $\DDD$ must themselves be lwqo, and at most one of $\mathcal{C}$ or $\mathcal{D}$ can contain arbitrarily long zigzag permutations. We also show that every class without long zigzag permutations has a growth rate which must be integral.


Introduction
Let C and D be permutation classes.The juxtaposition C D is the permutation class comprising all permutations formed by concatenations στ, where σ is order isomorphic to a permutation in C and τ is order isomorphic to a permutation in D.
A zigzag permutation (or just zigzag) is a permutation π = π(1) ¨¨¨π(n) with the property that there is no index i P [n ´2] such that π(i)π(i + 1)π(i + 2) forms a monotone increasing or decreasing pattern. 1 The main purpose of this note is to establish the following theorem.The juxtaposition of permutations was first introduced in Atkinson's foundational work [2], and has since been studied in terms of enumeration (see, for example, [7]) since it represents a natural yet non-trivial way to combine two permutation classes.Indeed, juxtapositions are a special case of grid classes, which we define in the next section.
The study of well-quasi-ordering and infinite antichains in permutation classes dates back to the 1970s in the work of Tarjan [16] and Pratt [14], and rose to prominence in the 2000s as a result of works such as Atkinson, Murphy and Ruškuc [3] and Murphy and Vatter [11].The stronger notion of labelled well-quasi-ordering dates back to Pouzet [13], but received little attention in the context of permutation classes until the current author's recent work with Vatter [8].
The rest of this paper is organised as follows.In Section 2 we briefly cover the requisite terminology.In Section 3 we provide a necessary and sufficient characterisation of permutation classes without long zigzags.As a by-product of this characterisation, we show that every permutation class without long zigzags has an integral growth rate.In Section 4 we prove that the juxtaposition of a labelled well-quasi-ordered permutation class with Av(21) or Av( 12) is again labelled well-quasi-ordered, and this, together with the characterisation from Section 3, enables us to complete our proof of Theorem 1.1.We finish with some concluding remarks in Section 5.

Preliminaries
Permutation classes We provide here only the minimum terminology required for our purposes, and refer the reader to [4] for fuller details.
A permutation of length n, typically denoted π = π(1) ¨¨¨π(n), is an ordering of the symbols in [n] = t1, . . ., nu.We say that σ = σ(1) ¨¨¨σ(k) is contained in π, and write σ ď π, if there exists a subsequence 1 ď i 1 ď ¨¨¨ď i k ď n such that the relative ordering of the points in π(i 1 ) ¨¨¨π(i k ) is the same as that of σ.That is, π contains a subsequence that is order isomorphic to σ.
A permutation class C is a set of permutations closed downwards under containment.Every such class can be described by its set of minimal forbidden elements, but for our purposes it suffices to record that Av One important family of permutation classes in the structural study of permutations are grid classes.These are defined by a gridding matrix M of permutation classes, and each permutation in Grid(M) has the property that its plot can be divided using horizontal and vertical lines into a grid of cells, of the same dimensions as M, and such that the entries in each cell of the plot are order isomorphic to a permutation that belongs to a class in the corresponding cell of M.
Of particular note are monotone grid classes, where each cell of M is Av(21), Av (12) or empty, and we say that a permutation class C is monotone griddable if it is the subclass of some monotone grid class.We need the following characterisation.The juxtaposition C D can alternatively be considered as Grid(M) where M = C D .Also of interest to us is the class of gridded permutations in a juxtaposition -denoted C | D -whose members comprise the permutations of C D together with a vertical line that witnesses the permutation's membership of the juxtaposition.Note that each permutation in C D can correspond to more than one gridded permutation in C | D. The same notion exists for grid classes defined by larger matrices: if C Ď Grid(M) then C 7 denotes the set of permutations in C equipped with horizontal and vertical lines to witness their membership of Grid(M).
Well-quasi-ordering A quasi-order (P, ď) is well-quasi-ordered (wqo) if it contains no infinite descending chain, and no infinite antichain -that is, a set of pairwise incomparable elements.
For quasi-ordered classes of combinatorial objects (such as permutation classes or gridded permutation classes), this condition typically reduces to checking for the presence of infinite antichains.
Given a quasi-order (P, ď), let P ˚denote the set of finite sequences of P. The set P ˚can be ordered using the generalised subword order: for v = v 1 ¨¨¨v m and w = w 1 ¨¨¨w n in P ˚, we say that v ĺ w if there exists a subsequence 1 ď i 1 ď ¨¨¨ď i m ď n such that v j ď w i j for all 1 ď j ď m.One celebrated result that we will need is Higman's lemma: Lemma 2.2 (Higman [9]).If (P, ď) is a wqo set, then so is (P ˚, ĺ).
Another way to combine wqo sets and obtain another wqo set is by taking products: ).Let (P, ď P ) and (Q, ď Q ) be wqo sets.Then P ˆQ is wqo under the product order, (p 1 , q 1 ) ď (p 2 , q 2 ) if and only if p 1 ď P p 2 and q The final piece of core wqo machinery we require is as follows.We say that a mapping Φ : P Ñ Q between two quasi-orders is order preserving if p 1 ď P p 2 implies Φ(p 1 ) ď Q Φ(p 2 ).We have: Proposition 2.4 (See [8, Proposition 1.10]).Let (P, ď P ) and (Q, ď Q ) be quasi-orders, and suppose that Φ : (P, Labelled well-quasi-ordering Let (L, ď L ) be any quasi-order.An L-labelling of a permutation π of length n (or of a gridded permutation of length n) is a mapping ℓ π from the indices of π to elements of L. We write the resulting L-labelled permutation as (π, ℓ π ), and the set of all L-labelled permutations from some set (or class) C is denoted C ≀ L.
The set C ≀ L induces a natural ordering: Let σ, π P C be of lengths m and n, respectively.We say that (σ, Finally, a set or class C is labelled well-quasi-ordered (lwqo) if C ≀ L is a wqo set for every wqo set (L, ď L ).We refer the reader to [8] for a complete treatment of lwqo in permutation classes.

Zigzags
A peak of a permutation π is a position i such that π(i ´1) ă π(i) ą π(i + 1).The peak set of π is The peak set has been much studied in enumerative and algebraic combinatorics, see, for example, Nyman [12] and Billey, Burdzy and Sagan [5], although here it simply provides convenient terminology to prove the following result.Proof.Suppose that the longest zigzag in C has length k.For any π P C of length n consider the peak set Peaks(π) and let i and j be two consecutive peaks (that is, there is no k P Peaks(π) such that i ă k ă j).Since there are no peaks between i and j, the sequence π(i) ¨¨¨π(j) must be a valley: that is, it is formed of a decreasing sequence, followed by an increasing sequence.Let v i be the index such that i ă v i ă j for which π(v i ) is minimal (the 'bottom of the valley').
Similarly, if ℓ is the leftmost peak in π, then π(1) ¨¨¨π(ℓ) is a valley, and if r is the rightmost peak in π, then π(r) ¨¨¨π(n) is a valley.In particular, we set v r to be the index in [r, n] for which π(v r ) is minimal.
Since the entries between consecutive peaks (and before the first, and after the last peak) form valleys, we see that the entries of π can be partitioned into a sequence of 2(|Peaks(π)| + 1) (possibly empty) intervals of entries, that alternately form decreasing and increasing permutations.By construction, the subpermutation formed on the indices Peaks(π) Y tv i : i P Peaks(π)u is a zigzag of length 2|Peaks(π)|.Thus, 2|Peaks(π)| ď k for every π P C, and hence π belongs to the grid class whose matrix is Av( 12) Av(21) Av( 12) Av(21) ¨¨¨Av( 12) Av( 21) Our next result establishes a more precise characterisation of classes without long zigzags.A vertical alternation is a permutation in which every odd-indexed entry lies above every evenindexed entry, or vice-versa.Some simple applications of the Erdős-Szekeres Theorem shows that every sufficiently long vertical alternation contains a long parallel or wedge alternation -see Figure 1.

Lemma 3.2. The permutation class C contains only finitely many zigzags if and only if C is monotone griddable and does not contain arbitrarily long vertical alternations.
Proof.If C is not monotone griddable then it contains À 21 or Á 12 by Theorem 2.1.In particular, for every n ě 1, C contains either 2143 ¨¨¨(2n)(2n ´1) or (2n ´1)(2n) ¨¨¨12, both of which Conversely, Proposition 3.1 shows that a class C with bounded length zigzags is contained in a monotone grid class comprising a single row, and this demonstrates both that C is monotone griddable and that it cannot contain arbitrarily long vertical alternations.
We finish this section by recording an interesting consequence of the above theorem.We need two auxiliary results.The first tells us that when a class is M-griddable, then it suffices to consider the upper and lower growth rates of the gridded permutations.The second result is attributed to Albert in one of Vatter's seminal works regarding the growth rates of permutation classes.
Proof of Corollary 3.3.By Proposition 3.1, we may suppose that C is contained in a monotone grid class Grid(M) whose defining matrix comprises a single row of (say) m cells.
The set C 7 of all M-gridded permutations in C is in bijection with a subword-closed language over an alphabet of size m (see, for example, the description in Section 7 of Vatter [18]), and in this bijection, the set of words corresponding to C is also subword-closed.By Proposition 3.5, the growth rate of C 7 exists and is integral, and thus by Proposition 3.4 the same is true of the growth rate of C.

Juxtapositions and lwqo
Since a class that contains only finitely many zigzags is M-griddable for a monotone grid class formed of a single row, we now want to understand what happens when we juxtapose an arbitrary lwqo class C with such a grid class.The bulk of the remaining work lies in the next theorem, which establishes that lwqo is preserved whenever we juxtapose an lwqo class with Av(21) or Av (12).Let (L, ď L ) be an arbitrary wqo set of labels.By Higman's lemma, (L ˚, ĺ) is wqo.Furthermore, by Proposition 2.3 the product L ˆL˚i s also wqo, and thus C ≀ (L ˆL˚) is wqo since C is lwqo.Finally, another application of Proposition 2.3 shows that C ≀ (L ˆL˚) ˆL˚i s wqo.
A typical element of C ≀ (L ˆL˚) ˆL˚h as the form P = ((π, k π ), z 1 ¨¨¨z q ) where π P C (of length n, say), z 1 , . . ., z q P L, where k π : [n] Ñ L ˆL˚i s given by for all i P [n], in which ℓ : [n] Ñ L, λ ij P L, and n i ě 0.
We now construct an order-preserving surjection Ψ from C≀(LˆL ˚)ˆL ˚to C | D≀L.This mapping takes an object P = ((π, k π ), z 1 ¨¨¨z q ) and outputs an L-labelled permutation in C | D≀L of length n + ř n i=1 n i + q.Specifically, in Ψ(P): • There are n points to the left of the gridline, order isomorphic to π.
• For i P [n], the ith point from the left is labelled by ℓ(i).
• There are ř n i=1 n i + q points to the right of the gridline, forming an increasing sequence.
• For i P [n], there are n i points to the right of the gridline that lie below the ith entry on the left, and above the next highest entry on the left (if this exists).These n i points are labelled λ i1 , . . ., λ in i from bottom to top.
• Above the highest entry on the left of the gridline, there are q points to the right of the gridline, labelled z 1 , . . ., z q from bottom to top.
See Figure 2. The proof will be completed by showing that Ψ is an order-preserving surjection.
First, any labelled gridded permutation in C | D ≀ L comprises a set of points to the left of the gridline (that form a permutation from C with labels from L), interleaved by sequences of points to the right of the gridline (that form an increasing permutation, also with labels from L).With λ 21 ℓ( 6) this in mind, for any specified element of C | D ≀ L it is straightforward to identify a suitable preimage in C ≀ (L ˆL˚) ˆL˚, which shows that Ψ is surjective.
Let σ have length m and π length n.Since σ ď π as labelled permutations, there exists a subsequence 1 ď i 1 ă ¨¨¨ă i m ď n such that π(i 1 ) ¨¨¨π(i m ) is order isomorphic to σ, and and λ j1 ¨¨¨λ jm j ĺ κ i j 1 ¨¨¨κ i j n i j in generalised subword order.Finally, we also require w 1 ¨¨¨w p ĺ z 1 ¨¨¨z q .
To complete the proof, we show that Ψ(S) ď Ψ(P) as L-labelled gridded permutations.
The points to the left of the gridline in Ψ(S) and Ψ(P) form the L-labelled permutations (σ, ℓ σ ) and (π, ℓ π ), respectively.The subsequence 1 ď i 1 ă ¨¨¨ă i m ď n witnesses both that σ ď π, and that ℓ σ (j) ď L ℓ π (i j ), and hence (σ, ℓ σ ) ď (π, ℓ π ).We now consider the points to the right of the gridline.In Ψ(S), for each j P [m] the points immediately below the entry on the left corresponding to σ(j) form an increasing sequence of length m j labelled by λ j1 , . . ., λ jm j .Similarly, in Ψ(P), the points immediately below the entry corresponding to π(i j ) form an increasing sequence of length n i j labelled by κ i j 1 , . . ., κ i j n i j .Since λ j1 ¨¨¨λ jm j ĺ κ i j 1 ¨¨¨κ i j n i j , we can embed these m j labelled points of Ψ(S) in the n i j labelled points of Ψ(P).
Finally, in Ψ(S), there are p labelled entries to the right of the gridline that lie above all entries to the left of the grid line.Since w 1 ¨¨¨w p ĺ z 1 ¨¨¨z q , these p entries can be embedded in the q entries of Ψ(P) in the top-right.We have now embedded every labelled entry of Ψ(S) in Ψ(P), and the proof is complete.
Our approach to resolve one direction of Theorem 1.1 will be to apply the preceding theorem iteratively.For the other direction, we appeal to pre-existing antichain constructions, which are succinctly summarised by the following theorem.
The cell graph of a matrix M is the graph whose vertices are t(i, j) : M ij ‰ ∅u (corresponding to the non-empty cells of M), and (i, j) " (k, ℓ) if and only if i = k or j = ℓ, and there are no non-empty cells between these M ij and M kℓ in their common row or column.
Theorem 4.2 (See Brignall [6,Theorem 1.1]).Let M be a gridding matrix where every non-empty cell is an infinite permutation class.Then Grid(M) is not well-quasi-ordered whenever the cell graph of M has a cycle, or a component containing two or more cells that are not monotone griddable.
Note that the 'cyclic' case of the above theorem is originally due to Murphy and Vatter [11].
Proof of Theorem where E 1 and E 2 are each either Av(21) or Av (12).In any case, the cell graph of M comprises a component containing two cells that are not monotone griddable (again by Theorem 2.1), and hence Grid(M) is not wqo by Theorem 4.2.(See Figure 3 (middle) for a typical antichain element in this case.) Finally for this direction, suppose that both C and D are monotone griddable, but both contain arbitrarily long vertical alternations.In this case, C D contains Grid(M) for a matrix M of the following form where E 1 , E 2 , E 3 and E 4 are each either Av(21) or Av (12).In any case, the cell graph of M comprises a component that is a cycle, so Grid(M) is once again not wqo by Theorem 4.2, and hence neither is C D. (See Figure 3 (right) for a typical antichain element in this case.) For the other direction, suppose (without loss of generality) that C is lwqo, and D contains only bounded length zigzags.By Proposition 3.1, there exists a single-row monotone grid class E such that D Ď E. We claim that C E is lwqo.
Write E = Grid(M) where M = E 1 E 2 ¨¨¨E k for classes E i each equal to Av(21) or Av (12) (1 ď i ď k).Let C 0 = C, and for 1 ď i ď k set Now C 0 = C is lwqo, and it follows by induction and Theorem 4.1 that C i = C i´1 E i is lwqo for each i = 1, . . ., k.In particular C k = C E is lwqo.The result now follows since C D Ď C E.

Concluding remarks
The methods and ideas in this note can almost certainly be adapted to a characterisation of lwqo in grid classes, although it would likely be technically and notationally awkward to do so.
A more interesting future direction is to consider lwqo in subclasses of these grid classes.For example, while the juxtaposition of Á 12 with À 21 contains the infinite antichain comprising elements of the form shown on the left of Figure 3, there exist subclasses of this juxtaposition that are lwqo.Individual cases such as this are relatively easy to characterise, but a general answer seems further out of reach.
Can a similar characterisation can be achieved for (unlabelled) wqo?Although the antichain elements depicted in Figure 3 use two labels, the proof of Theorem 1.1 in fact uses only unlabelled antichains, so aspects of this question already have an answer.However, if C is a wqobut-not-lwqo class, then it is sometimes possible to break wqo by juxtaposing C with the class containing just the singleton permutation, while in other cases, C must be juxtaposed with two entries.In general, we cannot hope to make progress on this question without a significantly deeper understanding of wqo in permutation classes.

Theorem 1 . 1 .
The juxtaposition C D is labelled-well-quasi-ordered if and only if both C and D are lwqo, and at least one of C or D contains only finitely many zigzag permutations.

Figure 1 :
Figure 1: From left to right: a vertical alternation, a parallel alternation, and a wedge alternation.

Proposition 3 . 4 (
Vatter[17, Proposition 2.1]).For a matrix of permutation classes M and a class C Ď Grid(M), the upper or lower growth rate of C is equal, respectively, to the upper or lower growth rate of C 7 .

Theorem 4 . 1 .
Let C be an arbitrary lwqo class, and let D be a monotone class.Then C D is lwqo.Proof.By symmetry, we can assume that D = Av(21).Furthermore, it suffices to show that the gridded permutations, C | D, are lwqo, since for any quasi-order L, the mapping Φ : C | D ≀ L Ñ C D ≀ L that removes the gridline is an order-preserving surjection, and thus by Proposition 2.4, if C | D ≀ L is wqo then so is C D ≀ L.

Figure 3 :
Figure 3: Typical labelled antichain elements arising in juxtaposition classes.Here, we may take L = t‚, ˝u to be an antichain of size 2.
(12)osition 3.1.LetC be a permutation class that contains only finitely many zigzags.Then C is contained in Grid(M) for a matrix M comprising one row, and in which each entry is Av(21) or Av(12).2 The growth rate of a permutation class C (or gridded permutation class C 7 ), if it exists, is lim nÑ∞ |C n |, where C n denotes the set of permutations in C of length n.The existence of the growth rate of a class in general depends upon whether the upper and lower growth rates coincide, that is, whether lim sup nÑ∞ n a n a |C n | = lim inf nÑ∞ n a |C n |.Corollary 3.3.Let C be a class that contains only finitely many zigzags.Then gr(C) exists and is integral.