The Weak Order on Pattern-Avoiding Permutations

The weak order on the symmetric group is a well-known partial order which is also a lattice. We consider subposets of the weak order consisting of permutations avoiding a single pattern, characterizing the patterns for which the subposet is a lattice. These patterns have only a single small ascent or descent. We prove that all patterns for which the subposet is a sublattice have length at most three.


Introduction
For any positive integer n, the weak order on the symmetric group S n is well known to be a lattice [7], [13].There are also some interesting subposets of the weak order which are lattices, some of which can be described using pattern avoidance.In particular, the Tamari lattice is isomorphic to the weak order on 312 avoiding permutations ( [4], 9.6.Theorem), [9].Additionally, the Boolean lattice is isomorphic to the weak order on 312 and 231 avoiding permutations ( [8], Lemme 4).Both the Tamari lattice and Boolean lattice are actually sublattices of the weak order.The Bruhat order on pattern avoiding permutations has also been studied [1].
In this article we consider the case of avoiding a single pattern.Our main result is the following.
Theorem 1. S n (τ ) is a lattice for all n if and only if 1. τ has exactly one descent, and its magnitude is at most two, or 2. τ has exactly one ascent, and its magnitude is at most two.If we insist that the join and meet operations in our subposet agree with those in S n , then it turns out that both (1) and ( 2) must be satisfied.These conditions are sufficient, and the result simplifies to the following.
Theorem 2. S n (τ ) is a sublattice of S n for all n if and only if τ is one of 1, 12, 21, 213, 132, 231, or 312.
The weak order on permutations avoiding any one of 213, 132, 231, or 312 is isomorphic to the Tamari lattice.The others are either empty or a single vertex.
In the course of proving these theorems, we find a somewhat surprising result.
Theorem 3. If τ is a pattern of length k 2, the following are equivalent: 1. S n (τ ) is a lattice for all n.
This paper is organized as follows.Section 2 contains some preliminaries about pattern avoidance, lattices, and the weak order.Section 3 contains the proofs of the main theorems.In Section 4, we give some results about S n (T ), where T contains multiple patterns.

Permutations
A permutation of n letters is a bijection π : [n] → [n], where [n] denotes {1, 2, . . ., n}.The set of all permutations of n letters is denoted S n .We write our permutations in one line notation as A descent in a permutation π occurs in position i if π(i) > π(i + 1).The magnitude of a descent in position i is π(i) − π(i + 1).Ascents and their magnitudes are defined analogously.An inversion is a pair (i, j) with i < j and π −1 (i) > π −1 (j).That is, in the one line notation for π, i appears to the right of j.We define the inversion set I(π) of π to be the set of all inversions in π.For example, π = 24513 has an ascent of magnitude 1 in position 2, ascents of magnitude 2 in positions 1 and 4, and a descent of magnitude 4 in position 3. Its inversion set is I(24513) = {(1, 2), (1,4), (1,5), (3,4), (3,5)}.

Pattern Avoidance
Let w be a word consisting of k distinct integers.The standardization of w, denoted st(w), is obtained by replacing the smallest entry of w with a 1, the second smallest with a 2, and so on.A permutation π ∈ S n is said to contain a pattern τ ∈ S k if there is a subsequence π ′ of π such that st(π ′ ) = τ .Otherwise, π is said to avoid τ .For example, the permutation π = 1423 contains the pattern 132 twice, since st(142) = 132 and st(143) = 132.The permutation 2134 avoids 132.For a set of patterns T , we will use S n (T ) to denote the permutations of length n which avoid all of the patterns in T .We will also use S n (T ) to denote the set S n (T ) together with its order relation induced from the weak order.For simplicity of notation, we omit the set brackets for a single pattern.

Lattices
We assume that the reader is familiar with the definitions of partially ordered sets (posets), covering relations, Hasse diagrams, and dual posets ( [11], Chapter 3), or [2].
A lattice L is a poset with the following property.For all x, y ∈ L, the set of upper bounds {z ∈ L | z x, z y} has a unique minimal element, called the join of x and y and denoted x ∨ y, and the set of lower bounds {z ∈ L | z x, z y} has a unique maximal element, called the meet of x and y and denoted x ∧ y.
One useful lemma which follows directly from the definition of a lattice is the following.
Lemma 4. Let L be a lattice, with u, v, x, y, z ∈ L. Then u ∨ v = z = x ∧ y if and only if {t ∈ L | u t, v t, t x, t y} = {z}.
A join semi-lattice L is a poset where every pair of elements x, y ∈ L has a join x ∨ y.A meet semi-lattice L is a poset where every pair of elements x, y ∈ L has a meet x ∧ y.L is a lattice if and only if it is both a join semi-lattice and a meet semi-lattice.We will make use of the following lemma.Lemma 5. ( [11], Proposition 3.3.1.)Let P be a finite meet semi-lattice with a unique maximal element.Then P is a lattice.
Given posets Q and P , where the elements of Q are a subset of the elements of P , we say that Q is an induced subposet of P if x y in Q if and only if x y in P .
In this paper we will be interested in showing that certain induced subposets are lattices.We are not assuming that the join and meet operations are the same in the induced subposet as in the larger lattice.When these operations do coincide, we use a stronger term.An induced subposet P of a lattice L is a join sublattice if x, y ∈ P implies x ∧ y ∈ P , where ∧ is taken in L.An induced subposet P of a lattice L is a meet sublattice the electronic journal of combinatorics 21(3) (2014), #P3.15 if x, y ∈ P implies x ∨ y ∈ P .P is a sublattice of L if P is both a join sublattice and a meet sublattice.
Our main tool for investigating induced subposets is the following technical result.
Lemma 6.Let L be a join semi-lattice and P an induced subposet.Suppose that for all v ∈ L, the set {v ′ ∈ P | v ′ v} has a unique maximal element or is the empty set.Then P is a join sublattice of L.
Proof.Let x, y ∈ P , and suppose z = x ∨ y in L. We want to show that z ∈ P .The set {z ′ ∈ P | z ′ z} is nonempty since it contains both x and y, so by assumption it has a unique maximal element z * .The element z * is in {z ′ ∈ P | z ′ z}, so we have z * z.
By the maximality of z * , we have x z * and y z * .But since z = x ∨ y, we also have z z * by the definition of join.Hence z = z * so z ∈ P .

Weak Order
For permutations σ, π ∈ S n , we define σ π if and only if I(σ) ⊆ I(π).It follows from the definition that π covers σ if and only if π is obtained from σ by an adjacent transposition which changes an ascent to a descent.Figure 1 includes the Hasse diagram of the weak order on S 3 .The weak order on S n is well known to be a lattice [7], [13].Throughout this paper, we will abuse notation and let S n denote the set of permutations of length n together with the order relation given by the weak order.There are some standard bijections of the weak order which will be useful.Let w 0 ∈ S n be the strictly decreasing permutation n(n − 1) • • • 21. (i) π → πw 0 and π → w 0 π are order reversing bijections of the weak order on S n .
(ii) π → w 0 πw 0 is an order preserving bijection of the weak order on S n .
It follows immediately that π avoids τ if and only if πw 0 avoids τ w 0 .The same is true for the other two bijections.Combined with Lemma 7, we get the following.Lemma 8. Let π ∈ S n (T ).
(i) π → πw 0 and π → w 0 π are order reversing bijections of the weak order on S n (T ).
(ii) π → w 0 πw 0 is an order preserving bijection of the weak order on S n (T ).

Terminology
Let us introduce some terminology to be used in this section.Given a poset P and elements σ, σ ′ , µ, µ ′ ∈ P such that µ covers both σ and σ ′ and µ ′ covers both σ and σ ′ , we say that σ, σ ′ , µ, and µ ′ form a butterfly in the Hasse diagram of P , and that P contains a butterfly in its Hasse diagram.We call µ and µ ′ the top elements of the butterfly and σ and σ ′ the bottom elements of the butterfly.See Figure 2. Note that any poset P which contains a butterfly in its Hasse diagram cannot be a lattice or semi-lattice, since the meet of the top elements and the join of the bottom elements of the butterfly do not exist.Proof.Suppose σ, σ ′ , µ, and µ ′ form a butterfly in the Hasse diagram of S n (τ ).Suppose also that τ ends with a descent in position k − 1.In this case, define Let us show that σ m , σ ′ m , µ m , and µ ′ m avoid τ .We know that τ ends with a descent and σ m , σ ′ m , µ m , and µ ′ m each end with an increasing sequence of their greatest (m − n) letters.So if any of these contains τ , the occurrence of τ would have to be in the first n letters.But these letters are one of σ, σ ′ , µ, or µ ′ , each of which avoids τ .Hence For the remaining case, suppose that τ ends with an ascent.Define An analogous argument shows that µ m , µ ′ m , σ m and σ ′ m form a butterfly in the Hasse diagram of S m (τ ).

Proofs
For the rest of this section, let τ be a pattern of length k, with k 2. We prove our main results using three intermediate propositions.
Proposition 10.Suppose τ has at least two ascents and at least two descents.Then S n (τ ) is not a meet semi-lattice or a join semi-lattice for n k.
By Lemma 9, it suffices to find a butterfly in the Hasse diagram of S k (τ ).Define permutations Notice that µ i is obtained from τ by changing the ascent in position a i to an descent, and that σ i is obtained from τ by changing the descent in position d i to an ascent.We have By our definition of the weak order on S k in terms of inversion sets, we have µ 1 ∧µ 2 = τ = σ 1 ∨ σ 2 , where the meet and join are taken in S k .By Lemma 4, τ is the only permutation in S k between µ 1 , µ 2 , σ 1 and σ 2 .Recall that τ has length k.A permutation π of length k contains a pattern τ of length k if and only if π = τ , so S k (τ ) = S k \ {τ } as sets.Then by the definition of an induced subposet, µ 1 , µ 2 , σ 1 and σ 2 form a butterfly in the Hasse diagram of S k (τ ).
Proposition 11.Suppose τ has exactly one descent, and its magnitude is at least 3. Then S n (τ ) is not a meet semi-lattice or a join semi-lattice for n k + 1.
Proof.Given such a pattern τ , we have a set of letters {a, b, c, d} which appear in τ , with a < b < c < d in the usual integer ordering, and a appearing immediately to the right of d in τ .That is, τ has a descent in position π −1 (d) of magnitude d − a 3. Since τ has exactly one descent, it can be decomposed into two increasing subwords of consecutive letters.One is τ (1) < τ (2) < • • • d and the other is a < • • • < τ (k).Therefore we can assume that a, b, c, and d appear consecutively in τ .There are four possibilities for τ : where τ init and τ end are the appropriate increasing subwords.In each case we want to find a butterfly in the Hasse diagram of S k+1 (τ ), which completes the proof by Lemma 9. Case (i).Choose τ ′ ∈ S k+1 such that First, we want to show that µ 1 , µ 2 , σ 1 , and σ 2 avoid τ .Since τ has length k, an occurrence of τ in one of these permutations would use all letters but one.Such an occurrence would start with an increasing sequence of |τ init | + 3 letters, followed by a descent.This is impossible in µ 1 , µ 2 and σ 2 .The letter after the descent in an occurrence of τ must be smaller than the previous three.This is impossible in σ 1 .Hence µ 1 , µ 2 , σ 1 , σ 2 ∈ S k+1 (τ ).
). Define the following permutations: We leave to the reader the argument that µ 1 , µ 2 , σ 1 , and σ 2 form a butterfly in the Hasse diagram of S k+1 (τ ).It is entirely analogous to Case (i).
Case (iii).We have τ = τ init cdab τ end with a < b < c < d.This is the same as in Case (ii), except b and c are switched.In Case (ii) we did not need to use the relation b < c, so the same argument works in this case.Case (iv).This case is equivalent to Case (i) by using the symmetry operation π → w 0 πw 0 described in Lemma 8.
Proposition 12. Suppose τ has exactly one descent, and its magnitude is at most two.Then S n (τ ) is a join sublattice of S n .
It is possible that π 0 contains τ , but π 0 does not have an occurrence of τ with π(i h+1 ) and π(i h ) forming the descent.Since π has only finitely many letters, we can repeat this process and eventually end at a permutation π * such that π * avoids τ , and every permutation π ′ which avoids τ and satisfies π ′ < π also satisfies π ′ π * .Now we can apply Lemma 6 to complete the proof.
We are now in a position to prove our main results.In fact, we have results about semi-lattices as well.
Theorem 13. S n (τ ) is a join sublattice of S n for all n if and only if 1. τ has exactly one descent, and its magnitude is at most two, or 2. τ has no descents.
Proof.Let τ be a pattern of length k.Proposition 12 shows that if τ satisfies condition (1), then S n (τ ) is a join sublattice of S n .Suppose that τ satisfies condition (2), so that τ = 12 • • • k.In this case π avoids τ if and only if π does not contain an increasing subsequence of k letters.If π does not contain an increasing subsequence of k letters and π ′ π, then π ′ does not contain an increasing subsequence of k letters since I(π) ⊆ I(π ′ ).Therefore π, σ ∈ S n (τ ) implies π ∨ σ ∈ S n (τ ).That is, S n (τ ) is a join sublattice.Now we want to show conditions (1) and ( 2) are necessary for S n (τ ) to be a join sublattice of S n .Propositions 10 and 11 rule out all cases, except when τ has exactly one ascent, that ascent has magnitude at most 2, and τ has at least two descents.Suppose τ is of this type.Then τ is the join of two permutations, each obtained by changing one of the descents to an ascent.But these two permutations are in S k (τ ), so it cannot be a join sublattice.
Theorem 14. S n (τ ) is a join semi-lattice for all n if and only if the electronic journal of combinatorics 21(3) (2014), #P3.15 1. τ has exactly one descent, and its magnitude is at most two, or 2. τ has exactly one ascent, and its magnitude is at most two, or 3. τ has no descents.
Suppose τ has exactly one ascent, which is of magnitude at most two.Then by using π → πw 0 of Lemma 8 and Proposition 12, the dual poset to S n (τ ) is a join semi-lattice.That is, S n (τ ) is a meet semi-lattice.Since n(n − 1) • • • 21 avoids τ , S n (τ ) has a maximal element.By the dual to Lemma 5, S n (τ ) is a lattice, and hence a join semi-lattice.
Using the map π → πw 0 of Lemma 8, we have dual results.
Theorem 15. S n (τ ) is a meet sublattice of S n for all n if and only if 1. τ has exactly one ascent, and its magnitude is at most two, or 2. τ has no ascents.
Theorem 16. S n (τ ) is a meet semi-lattice for all n if and only if 1. τ has exactly one ascent, and its magnitude is at most two, or 2. τ has exactly one descent, and its magnitude is at most two, or 3. τ has no ascents.
Since S n (τ ) is a lattice if and only if it is both a meet semi-lattice and a join semilattice, we have proven Theorem 1.We have also shown that the only τ for which S n (τ ) is a semi-lattice but not a lattice are the strictly increasing and strictly decreasing patterns.
We can apply the same reasoning to sublattices, and get Theorem 2. Propositions 10, 11, and 12 also immediately imply Theorem 3.

Multiple patterns
Theorem 1 does not generalize immediately to larger sets of patterns.In particular, even if both S n (τ 1 ) and S n (τ 2 ) are lattices, S n ({τ 1 , τ 2 }) is not necessarily a lattice.Moreover, it is not necessary for both S n (τ 1 ) and S n (τ 2 ) to be lattices in order for S n ({τ 1 , τ 2 }) to be a lattice.For example, consider S n ({2134, 2143}), which is one case of the following result.Proof.Observe that S n (T ) is the set of permutations such that for each descent π(i) > π(i + 1), we have |{j | j > i, π(j) > π(i)}| < k − 2. So if π ∈ S n \ S n (T ), then there is a unique maximal element π ′ ∈ S n (T ) less than π.The element π ′ is obtained by changing all descents which violate the condition above to ascents.Therefore S n (T ) is a join semi-lattice by Lemma 6.We also have 12 • • • n ∈ S n (T ), so S n (T ) is a lattice by the dual statement of Lemma 5.

Figure 2 :
Figure 2: A butterfly in a Hasse diagram.

Lemma 9 .
If the Hasse diagram of S n (τ ) contains a butterfly, then so does the Hasse diagram of S m (τ ) for m n.