Shuffle-compatible permutation statistics II: the exterior peak set

This is a continuation of arXiv:1706.00750 by Gessel and Zhuang (but can be read independently from the latter). We study the shuffle-compatibility of permutation statistics -- a concept introduced in arXiv:1706.00750, although various instances of it have appeared throughout the literature before. We prove that (as Gessel and Zhuang have conjectured) the exterior peak set statistic (Epk) is shuffle-compatible. We furthermore introduce the concept of an"LR-shuffle-compatible"statistic, which is stronger than shuffle-compatibility. We prove that Epk and a few other statistics are LR-shuffle-compatible. Furthermore, we connect these concepts with the quasisymmetric functions, in particular the dendriform structure on them.


Remark on alternative versions
This paper also has a detailed version [Grinbe18], which includes some proofs that have been omitted from the present version as well as more details on some other proofs and further results in Sections 4, 5 and 6.

Notations and definitions
Let us first introduce the definitions and notations that we will use in the rest of this paper. Many of these definitions appear in [GesZhu17] already; we have tried to deviate from the notations of [GesZhu17] as little as possible.
(h) The right peak set of π is defined to be the set of all right peaks of π. This set is denoted by Rpk π, and is always a subset of {2, 3, . . . , n}. It is easy to see that (for n ≥ 2) we have Rpk π = Pk π ∪ {n | π n−1 < π n } .
(j) The exterior peak set of π is defined to be the set of all exterior peaks of π. This set is denoted by Epk π, and is always a subset of [n]. It is easy to see that (for n ≥ 2) we have Epk π = Pk π ∪ {1 | π 1 > π 2 } ∪ {n | π n−1 < π n } = Lpk π ∪ Rpk π (where, again, {n | π n−1 < π n } is the 1-element set {n} if π n−1 < π n , and otherwise is the empty set).

Shuffles and shuffle-compatibility
Definition 1.10. Let π and σ be two permutations.
(a) We say that π and σ are disjoint if no letter appears in both π and σ. (b) Assume that π and σ are disjoint. Set m = |π| and n = |σ|. Let τ be an (m + n)-permutation. Then, we say that τ is a shuffle of π and σ if both π and σ are subsequences of τ.
(c) We let S (π, σ) be the set of all shuffles of π and σ.
If π and σ are two disjoint permutations, and if τ is a shuffle of π and σ, then each letter of τ must be either a letter of π or a letter of σ. (This follows easily from the pigeonhole principle.) If π and σ are two disjoint permutations, then S (π, σ) = S (σ, π) is an m + n m element set, where m = |π| and n = |σ|. Definition 1.10 (b) is used, e.g., in [Greene88]. From the point of view of combinatorics on words, it is somewhat naive, as it fails to properly generalize to the case when the words π and σ are no longer disjoint 1 . But we will not be considering this general case, since our results do not seem to straightforwardly extend to it (although we might have to look more closely); thus, Definition 1.10 will suffice for us. Definition 1.11. (a) If a 1 , a 2 , . . . , a k are finitely many arbitrary objects, then {a 1 , a 2 , . . . , a k } multi denotes the multiset whose elements are a 1 , a 2 , . . . , a k (each appearing with the multiplicity with which it appears in the list (a 1 , a 2 , . . . , a k )).
In other words, a permutation statistic st is shuffle-compatible if and only if it has the following property: • If π and σ are two disjoint permutations, and if π ′ and σ ′ are two disjoint permutations, and if these permutations satisfy The notion of a shuffle-compatible permutation statistic was coined by Gessel and Zhuang in [GesZhu17], where various statistics were analyzed for their shuffle-compatibility. In particular, it was shown in [GesZhu17] that the statistics Des, Pk, Lpk and Rpk are shuffle-compatible. Our next goal is to prove the same for the statistic Epk.

Extending enriched P-partitions and the exterior peak set
We are going to define Z-enriched P-partitions, which are a straightforward generalization of the notions of "P-partitions" [Stanle72], "enriched P-partitions" [Stembr97,§2] and "left enriched P-partitions" [Peters05]. We will then consider a new particular case of this notion, which leads to a proof of the shufflecompatibility of Epk conjectured in [GesZhu17] (Theorem 2.48 below). We remark that Bruce Sagan and Duff Baker-Jarvis are currently working on an alternative, bijective approach to the shuffle-compatibility of permutation statistics, which may lead to a different proof of this fact.

Lacunar sets
First, let us briefly study lacunar sets, a class of subsets of Z that are closely connected to exterior peaks. We start with the definition: Definition 2.1. A set S of integers is said to be lacunar if each s ∈ S satisfies s + 1 / ∈ S.
In other words, a set of integers is lacunar if and only if it contains no two consecutive integers. For example, the set {2, 5, 7} is lacunar, while the set {2, 5, 6} is not.
Lacunar sets of integers are also called sparse sets in some of the literature (though the latter word has several competing meanings).
Definition 2.2. Let n ∈ N. We define a set L n of subsets of [n] as follows: • If n is positive, then L n shall mean the set of all nonempty lacunar subsets of [n].
• If n = 0, then L n shall mean the set {∅}.
For example, Proof of Proposition 2.3. Recall that L n is the set of all nonempty lacunar subsets of [n] (since n is positive). Thus, |L n | is the number of all lacunar subsets of [n] minus 1 (since the empty set ∅, which is clearly a lacunar subset of [n], is withheld from the count). But a known fact (see, e.g., [Stanle11, Exercise 1.35 a.]) says that the number of lacunar subsets of [n] is f n+2 . Combining the preceding two sentences, we conclude that |L n | = f n+2 − 1. This proves Proposition 2.3.
The following observation is easy: Proposition 2.4. Let n ∈ N. Let π be an n-permutation. Then, Epk π ∈ L n .
Proof of Proposition 2.4. If n = 0, then the statement is obvious (since in this case, we have Epk π = ∅ ∈ L 0 ). Thus, WLOG assume that n = 0. Hence, n is positive. Hence, L n is the set of all nonempty lacunar subsets of [n] (by the definition of L n ).
The set Epk π is lacunar (since two consecutive integers cannot both be exterior peaks of π), and is also nonempty (since π −1 (n) is an exterior peak of π). Therefore, Epk π is a nonempty lacunar subset of [n]. In other words, Epk π ∈ L n (since L n is the set of all nonempty lacunar subsets of [n]). This proves Proposition 2.4. Proposition 2.4 actually has a sort of converse: Proposition 2.5. Let n ∈ N. Let Λ be a subset of [n]. Then, there exists an n-permutation π satisfying Λ = Epk π if and only if Λ ∈ L n .
Proof of Proposition 2.5. Omitted; see [Grinbe18] for a proof.
Next, let us introduce a total order on the finite subsets of Z: Definition 2.6. (a) Let P be the set of all finite subsets of Z.
(b) If A and B are any two sets, then A △ B shall denote the symmetric difference of A and B. This is the set (A ∪ B) \ (A ∩ B) = (A \ B) ∪ (B \ A). It is well-known that the binary operation △ on sets is associative.
If A and B are two distinct sets, then the set A △ B is nonempty. Also, if A ∈ P and B ∈ P, then A △ B ∈ P. Thus, if A and B are two distinct sets in P, then min (A △ B) ∈ Z is well-defined.
(c) We define a binary relation < on P as follows: For any A ∈ P and B ∈ P, we let A < B if and only if A = B and min (A △ B) ∈ A. (This definition makes sense, because the condition Note that this relation < is similar to the relation < in [AgBeNy03, Lemma 4.3].
Proposition 2.7. The relation < on P is the smaller relation of a total order on P.
In the following, we shall regard the set P as a totally ordered set, equipped with the order from Proposition 2.7. Thus, for example, two sets A and B in P satisfy A ≥ B if and only if either A = B or B < A.
Definition 2.8. Let S be a subset of Z. Then, we define a new subset S + 1 of Z by setting For example, {2, 5} + 1 = {3, 6}. Note that a subset S of Z is lacunar if and only if S ∩ (S + 1) = ∅. Proposition 2.9. Let Λ ∈ P and R ∈ P be such that the set R is lacunar and R ⊆ Λ ∪ (Λ + 1). Then, R ≥ Λ (with respect to the total order on P).
Proof of Proposition 2.9. Assume the contrary. Thus, R < Λ (since P is totally ordered). In other words, R = Λ and min (R △ Λ) ∈ R (by the definition of the relation <).

Z-enriched (P, γ)-partitions
Convention 2.10. By abuse of notation, we will often use the same notation for a poset P = (X, ≤) and its ground set X when there is no danger of confusion. In particular, if x is some object, then "x ∈ P" shall mean "x ∈ X". Definition 2.11. A labeled poset means a pair (P, γ) consisting of a finite poset P = (X, ≤) and an injective map γ : X → A for some totally ordered set A. The injective map γ is called the labeling of the labeled poset (P, γ). The poset P is called the ground poset of the labeled poset (P, γ).
Convention 2.12. Let N be a totally ordered set, whose (strict) order relation will be denoted by ≺. Let + and − be two distinct symbols. Let Z be a subset of the set N × {+, −}. For each q = (n, s) ∈ Z, we denote the element n ∈ N by |q|, and we call the element s ∈ {+, −} the sign of q. If n ∈ N , then we will denote the two elements (n, +) and (n, −) of N × {+, −} by +n and −n, respectively.
We equip the set Z with a total order, whose (strict) order relation ≺ is defined by (n, s) ≺ n ′ , s ′ if and only if either n ≺ n ′ or n = n ′ and s = − and s ′ = + . Let Pow N be the ring of all formal power series over Q in the indeterminates x n for n ∈ N .
We fix N and Z throughout Subsection 2.2. That is, any result in this subsection is tacitly understood to begin with "Let N be a totally ordered set, whose (strict) order relation will be denoted by ≺, and let Z be a subset of the set N × {+, −}"; and the notations of this convention shall always be in place throughout this Subsection.
Whenever ≺ denotes some strict order, the corresponding weak order will be denoted by . (Thus, a b means "a ≺ b or a = b".) Definition 2.13. Let (P, γ) be a labeled poset. A Z-enriched (P, γ)-partition means a map f : P → Z such that for all x < y in P, the following conditions hold: (Of course, this concept depends on N and Z, but these will always be clear from the context.) Example 2.14. Let P be the poset with the following Hasse diagram: (that is, the ground set of P is {a, b, c, d}, and its order relation is given by a < c < b and a < d < b). Let γ : P → Z be a map that satisfies γ (a) < γ (b) < γ (c) < γ (d) (for example, γ could be the map that sends a, b, c, d to 2, 3, 5, 7, respectively). Then, (P, γ) is a labeled poset. A Z-enriched (P, γ)partition is a map f : P → Z satisfying the following conditions: For example, if N = P (the totally ordered set of positive integers, with its usual ordering) and Z = N × {+, −}, then the map f : P → Z sending a, b, c, d to +2, −3, +2, −3 (respectively) is a Z-enriched (P, γ)-partition. Notice that the total ordering on Z in this case is given by rather than by the familiar total order on Z.
The concept of a "Z-enriched (P, γ)-partition" generalizes three notions in existing literature: that of a "(P, γ)-partition", that of an "enriched (P, γ)-partition", and that of a "left enriched (P, γ)-partition" 2 : Definition 2.17. Let P be any finite poset. Then, L (P) shall denote the set of all linear extensions of P. A linear extension of P shall be understood simultaneously as a totally ordered set extending P and as a list (w 1 , w 2 , . . . , w n ) of all elements of P such that no two integers i < j satisfy w i ≥ w j in P. 2 The ideas behind these three concepts are due to Stanley [Stanle72], Stembridge [Stembr97, §2] and Petersen [Peters05], respectively, but the precise definitions are not standardized across the literature. We define a "(P, γ)-partition" as in [Stembr97, §1.1]; this definition differs noticeably from Stanley's (in particular, , but the differences do not end here). We define an "enriched (P, γ)-partition" as in [Stembr97,§2]. Finally, we define a "left enriched (P, γ)-partition" to be a Z-enriched (P, γ)-partition where N = N and Z = (N × {+, −}) \ {−0}; this definition is equivalent to Petersen's [Peters06, Definition 3.4.1] up to some differences of notation (in particular, Petersen assumes that the ground set of P is already a subset of P, and that the labeling γ is the canonical inclusion map P → P; also, he identifies the elements +0, −1, +1, −2, +2, . . . of (N × {+, −}) \ {−0} with the integers 0, −1, +1, −2, +2, . . ., respectively). Note that the definition Petersen Proof of Proposition 2.18. This is analogous to the proof of [Stembr97, Lemma 2.1]. See [Grinbe18] for details.
Definition 2.19. Let (P, γ) be a labeled poset. We define a power series This is easily seen to be convergent in the usual topology on Pow N . (Indeed, for every monomial m in Pow N , there exist at most |P|! · 2 |P| many f ∈ Corollary 2.20. For any labeled poset (P, γ), we have Proof of Corollary 2.20. Follows straight from Proposition 2.18. Definition 2.21. Let P be any set. Let A be a totally ordered set. Let γ : P → A and δ : P → A be two maps. We say that γ and δ are order-isomorphic if the following holds: For every pair (p, q) ∈ P × P, we have γ (p) ≤ γ (q) if and only if δ (p) ≤ δ (q). Lemma 2.22. Let (P, α) and (P, β) be two labeled posets with the same ground poset P. Assume that the maps α and β are order-isomorphic. Then: Proof of Lemma 2.22. (a) If x and y are two elements of P, then we have the following equivalences: (Indeed, the first of these equivalences holds because α and β are order-isomorphic; the second is the contrapositive of the first; the third is obtained from the second by swapping x with y.) Hence, the conditions "α (x) > α (y)" and "α (x) < α (y)" in the definition of a Z-enriched (P, α)-partition are equivalent to the conditions "β (x) > β (y)" and "β (x) < β (y)" in the definition of a Z-enriched (P, β)-partition. Therefore, the Z-enriched (P, α)-partitions are precisely the Z-enriched (P, β)-partitions. In other words, E (P, α) = E (P, β). This proves Lemma 2.22 . This set is denoted by P ⊔ Q, and comes with two canonical injections The images of these two injections are disjoint, and their union is P ⊔ Q.
If f : P ⊔ Q → X is any map, then the restriction of f to P is understood to be the map f • ι 0 : P → X, whereas the restriction of f to Q is understood to be the map f • ι 1 : Q → X. (Of course, this notation is ambiguous when P = Q.) When the sets P and Q are already disjoint, it is common to identify their disjoint union P ⊔ Q with their union P ∪ Q via the map Under this identification, the restriction of a map f : P ⊔ Q → X to P becomes identical with the (literal) restriction f | P of the map f : P ∪ Q → X (and similarly for the restrictions to Q). (b) Let P and Q be two posets. The disjoint union of the posets P and Q is the poset P ⊔ Q whose ground set is the disjoint union P ⊔ Q, and whose order relation is defined by the following rules: • If p and p ′ are two elements of P, then (0, p) < (0, p ′ ) in P ⊔ Q if and only if p < p ′ in P.
• If q and q ′ are two elements of Q, then (1, q) < (1, q ′ ) in P ⊔ Q if and only if q < q ′ in Q.
• If p ∈ P and q ∈ Q, then the elements (0, p) and (1, q) of P ⊔ Q are incomparable.
Proposition 2.24. Let (P, γ) and (Q, δ) be two labeled posets. Let (P ⊔ Q, ε) be a labeled poset whose ground poset P ⊔ Q is the disjoint union of P and Q, and whose labeling ε is such that the restriction of ε to P is order-isomorphic to γ and such that the restriction of ε to Q is order-isomorphic to δ. Then, Proof of Proposition 2.24. We WLOG assume that the ground sets P and Q are disjoint; thus, we can identify P ⊔ Q with the union P ∪ Q. Let us make this identification.
The restriction ε | P of ε to P is order-isomorphic to γ. Hence, Lemma 2.22 (a) (applied to α = ε | P and β = γ) yields E (P, ε | P ) = E (P, γ). Similarly, It is easy to see that a map f : Therefore, the map is a bijection (this is easy to see). In other words, the map is a bijection (since E (P, ε | P ) = E (P, γ) and E Q, ε | Q = E (Q, δ)). Now, the definition of Γ Z (P ⊔ Q, ε) yields This proves Proposition 2.24.
Let us recall the concept of a "poset homomorphism": Definition 2.26. Let P and Q be two posets. A map f : P → Q is said to be a poset homomorphism if for any two elements x and y of P satisfying x ≤ y in P, It is well-known that if U and V are any two finite totally ordered sets of the same size, then there is a unique poset isomorphism U → V. Thus, if w is a finite totally ordered set with n elements, then there is a unique poset isomorphism w → [n]. Now, we claim the following: Proposition 2.27. Let w be a finite totally ordered set with ground set W. Let n = |W|. Let w be the unique poset isomorphism w → [n]. Let γ : W → {1, 2, 3, . . .} be any injective map. Then, Proof of Proposition 2.27. Clearly, (w, γ) is a labeled poset (since γ is injective). The map γ • w −1 : [n] → {1, 2, 3, . . .} is an injective map, thus an n-permutation. Hence, Γ Z γ • w −1 is well-defined, and its definition yields , and thus is an isomorphism of labeled posets 3 from (w, γ) to [n] , γ • w −1 . Hence, is a bijection (since any isomorphism of labeled posets induces a bijection between their Z-enriched (P, γ)-partitions). Furthermore, it satisfies ∏ For the following corollary, let us recall that a bijective poset homomorphism is not necessarily an isomorphism of posets (since its inverse may and may not be a poset homomorphism).
Corollary 2.28. Let (P, γ) be a labeled poset. Let n = |P|. Then, Proof of Corollary 2.28. For each totally ordered set w with ground set P, we let w be the unique poset isomorphism w → [n]. If w is a linear extension of P, then this map w is also a bijective poset homomorphism P → [n] (since every poset homomorphism w → [n] is also a poset homomorphism P → [n]). Thus, for each w ∈ L (P), we have defined a bijective poset homomorphism w : P → [n]. We thus have defined a map L (P) → {bijective poset homomorphisms P → [n]} , (2) This map is injective (indeed, a linear extension w ∈ L (P) can be uniquely reconstructed from w) and surjective (because if x is a bijective poset homomorphism P → [n], then the linear extension w ∈ L (P) defined (as a list) by Hence, this map is a bijection. Corollary 2.20 yields But recall that the map (2) is a bijection. Thus, we can substitute x for w in the sum ∑ w∈L(P) This proves Corollary 2.28.
Corollary 2.29. Let n ∈ N and m ∈ N. Let π be an n-permutation and let σ be an m-permutation such that π and σ are disjoint. Then, . .} whose restriction to [n] is π and whose restriction to [m] is σ. This map ε is injective, since π and σ are disjoint permutations. Thus, ([n] ⊔ [m] , ε) is a labeled poset.
The following two observations are easy to show (see [Grinbe18] for detailed proofs): Observation 2: If τ ∈ S (π, σ), then there exists a unique bijective poset homomorphism x : is well-defined (by Observation 1) and is a bijection (by Observation 2). Hence, we can substitute ε • x −1 for τ in the sum ∑ τ∈S(π,σ) Γ Z (τ). We thus obtain bijective poset homomorphism Multiplying these two equalities, we obtain This proves Corollary 2.29.

Exterior peaks
So far we have been doing general nonsense. Let us now specialize to a situation that is connected to exterior peaks.
Convention 2.30. From now on, we set N = {0, 1, 2, . . .} ∪ {∞}, with total order given by 0 ≺ 1 ≺ 2 ≺ · · · ≺ ∞, and we set Recall that the total order on Z has Definition 2.31. Let S be a subset of Z. A map χ from S to a totally ordered set K is said to be V-shaped if there exists some t ∈ S such that the map χ | {s∈S | s≤t} is strictly decreasing while the map χ | {s∈S | s≥t} is strictly increasing. Notice that this t ∈ S is uniquely determined in this case; namely, it is the unique k ∈ S that minimizes χ (k).
Thus, roughly speaking, a map from a subset of Z to a totally ordered set is V-shaped if and only if it is strictly decreasing up until a certain value of its argument, and then strictly increasing afterwards. For example, the 6-permutation (5, 1, 2, 3, 4) is V-shaped (keep in mind that we regard n-permutations as injective maps [n] → P), whereas the 4-permutation (3, 1, 4, 2) is not.
Using this definition, we can rewrite the definition of Γ Z (π) as follows: Proposition 2.33. Let n ∈ N. Let π be any n-permutation. Then, Proof of Proposition 2.33. Easy consequence of the definitions (see [Grinbe18] for details).
Definition 2.34. Let n ∈ N. Let g : [n] → N be any map. Let π be an n-permutation. We shall say that g is π-amenable if it has the following properties: (iii') The map π | g −1 (∞) is strictly decreasing. (This allows the case when g −1 (∞) = ∅.) (iv') The map g is weakly increasing.
Proposition 2.35. Let n ∈ N. Let π be any n-permutation. Then, Proof of Proposition 2.35 (sketched). The claim will immediately follow from (5) once we have shown the following two observations: It thus remains to prove these two observations. Let us do this: , the following conditions hold: (This is due to the definition of a Z-enriched ([n] , π)-partition.) Condition (i) shows that the map f is weakly increasing. Condition (ii) shows that for each h ∈ N , the map π | f −1 (+h) is strictly increasing. Condition (iii) shows that for each h ∈ N , the map )), and can be written as the union of its two disjoint subsets f −1 (+h) and f −1 (−h). Furthermore, each element of f −1 (−h) is smaller than each element of f −1 (+h) (since f is weakly increasing), and we know that the map π | f −1 (−h) is strictly decreasing while the map π | f −1 (+h) is strictly increasing. Hence, the map π | g −1 (h) is strictly decreasing up until some value of its argument (namely, either the largest element of f −1 (−h), or the smallest element of f −1 (+h), depending on which of these two elements has the smaller image under π), and then strictly increasing from there on. In other words, the map π | g −1 (h) is V-shaped. Thus, Property (ii') in Definition 2.34 holds. Finally, Property (iv') in Definition 2.34 holds because f is weakly increasing. We have thus checked all four properties in Definition 2.34; thus, g is π-amenable. In other words, We are wondering to what extent the map f is determined by g and π.
Everything that we said in the proof of Observation 1 still holds in our situation (since g = | f |).
In order to determine the map f , it clearly suffices to determine the sets f −1 (q) for all q ∈ Z. In other words, it suffices to determine the set f −1 (+0), the set Recall from the proof of Observation 1 that then this uniquely determines f −1 (+h) and f −1 (−h). Thus, we focus only on the case when g −1 (h) = ∅.
The map g is weakly increasing (by Property (iv') in Definition 2.34). Hence, As in the proof of Observation 1, we can see that each element of Consider this β. Clearly, f −1 (−h) and f −1 (+h) are uniquely determined by β; we just need to find out which values β can take.
As in the proof of Observation 1, we can see that the map π | f −1 (−h) is strictly decreasing while the map π | f −1 (+h) is strictly increasing. Let k be the element of g −1 (h) minimizing π (k). Then, the map π is strictly decreasing on the set The map π | f −1 (−h) is strictly decreasing. In other words, the map π is strictly decreasing on the set f −1 (−h) = [α, β]. On the other hand, the map π is strictly increasing on the set u ∈ g −1 (h) | u ≥ k . Hence, the two sets [α, β] and u ∈ g −1 (h) | u ≥ k cannot have more than one element in common (since π is strictly decreasing on one and strictly increasing on the other). Thus, k ≥ β.
A similar argument shows that k ≤ β + 1. Combining these inequalities, we obtain k ∈ {β, β + 1}, so that β ∈ {k, k − 1}. This shows that β can take only two values: k and k − 1. Now, let us forget that we fixed h. We have shown that for each h ∈ g ([n]) ∩ {1, 2, 3, . . .}, the sets f −1 (+h) and f −1 (−h) are uniquely determined once the integer β is chosen, and that this integer β can be chosen in two ways.
Working the above argument backwards, we see that each way of making these decisions actually leads to a map f ∈ Now, let us observe that if g : [n] → N is a weakly increasing map (for some n ∈ N), then the fibers of g (that is, the subsets g −1 (h) of [n] for various h ∈ N ) are intervals of [n] (possibly empty). Of course, when these fibers are nonempty, they have smallest elements and largest elements. We shall next study these elements more closely.
Definition 2.36. Let n ∈ N. Let g : [n] → N be any map. We define a subset FE (g) of [n] as follows: In other words, FE (g) is the set comprising the smallest elements of all nonempty fibers of g except for g −1 (0) as well as the largest elements of all nonempty fibers of g except for g −1 (∞). We shall refer to the elements of FE (g) as the fiber-ends of g.
Lemma 2.37. Let n ∈ N. Let Λ ∈ L n . Then, there exists a weakly increasing map g : Proof of Lemma 2.37 (sketched). If n = 0, then Lemma 2.37 holds for obvious reasons. Thus, WLOG assume that n = 0. Hence, n is a positive integer. Thus, L n is the set of all nonempty lacunar subsets of [n] (by the definition of L n ). Therefore, from Λ ∈ L n , we conclude that Λ is a nonempty lacunar subset of [n].
Proposition 2.38. Let n ∈ N. Let π be an n-permutation. Let g : [n] → N be any weakly increasing map. Then, the map g is π-amenable if and only if Epk π ⊆ FE (g).
Proof of Proposition 2.38. The map g is weakly increasing. Thus, all nonempty fibers g −1 (h) of g are intervals of [n]. Recall that g is π-amenable if and only if the four Properties (i'), (ii'), (iii') and (iv') in Definition 2.34 hold. Consider these four properties. Since Property (iv') automatically holds (since we assumed g to be weakly increasing), we thus only need to discuss the other three: • Property (i') is equivalent to the statement that every exterior peak of π that lies in the fiber g −1 (0) must be the largest element of this fiber. (Indeed, a strictly increasing map is characterized by having no exterior peaks except for the largest element of its domain.) • Property (ii') is equivalent to the statement that every peak of π that lies in a fiber g −1 (h) with h ∈ g ([n]) ∩ {1, 2, 3, . . .} must be either the smallest or the largest element of this fiber (because the restriction π | g −1 (h) is V-shaped if and only if no peak of π appears in the interior of this fiber 5 ). Moreover, we can replace the word "peak" by "exterior peak" in this sentence (since the exterior peaks 1 and n must automatically be the smallest and the largest element of whatever fibers they belong to).
• Property (iii') is equivalent to the statement that every exterior peak of π that lies in the fiber g −1 (∞) must be the smallest element of this fiber.
(Indeed, a strictly decreasing map is characterized by having no exterior peaks except for the smallest element of its domain.) Combining all of these insights, we conclude that the four Properties (i'), (ii'), (iii') and (iv') hold if and only if every exterior peak of π is a fiber-end of g. In other words, g is π-amenable if and only if every exterior peak of π is a fiber-end of g (since g is π-amenable if and only if the four Properties (i'), (ii'), (iii') and (iv') hold). In other words, g is π-amenable if and only if Epk π ⊆ FE (g). This proves Proposition 2.38.
In other words, Proof of Corollary 2.42. From (8), we obtain Γ Z (π) = K Z n,Epk π . Similarly, Γ Z (σ) = K Z m,Epk σ . Multiplying these two equalities, we obtain Γ Z (π) · Γ Z (σ) = K Z n,Epk π · K Z m,Epk σ . Hence, Proof of Lemma 2.46. (a) A weakly increasing map g : [n] → N can be uniquely reconstructed from the multiset {g (1) , g (2) , . . . , g (n)} multi of its values (because it is weakly increasing, so there is only one way in which these values can be ordered). Hence, a weakly increasing map g : [n] → N can be uniquely reconstructed from the monomial x g (since this monomial x g = x g(1) x g(2) · · · x g(n) encodes the multiset {g (1) , g (2) , . . . , g (n)} multi (since x h and x g are two monomials) First proof of Proposition 2.44. Recall Definition 2.6. In the following, we shall regard the set P as a totally ordered set, equipped with the order from Proposition 2.7. Clearly, L n ⊆ P. Hence, we consider L n as a totally ordered set, whose total order is inherited from P.
Let (a R ) R∈L n ∈ Q L n be a family of scalars (in Q) such that ∑ R∈L n a R K Z n,R = 0. We are going to show that (a R ) R∈L n = (0) R∈L n .
Indeed, assume the contrary. Thus, (a R ) R∈L n = (0) R∈L n . Hence, there exists some R ∈ L n such that a R = 0. Let Λ be the largest such R (with respect to the total order on L n we have introduced above). Hence, Λ is an element of L n and satisfies a Λ = 0; but every element R ∈ L n satisfying R > Λ must satisfy Lemma 2.37 shows that there exists a weakly increasing map g : For every R ∈ L n satisfying R = Λ, we have [Proof of (11): Let R ∈ L n be such that R = Λ. We must prove (11). Assume the contrary. Thus, x g a R K Z n,R = 0. In other words, a R x g K Z n,R = 0. Hence, a R = 0 and x g K Z n,R = 0. From the definition of L n , it follows easily that every element of L n is a lacunar subset of [n]. Hence, R is a lacunar subset of [n] (since R ∈ L n ).
But Lemma 2.46 (b) (applied to h = g) yields Hence, Thus, Proposition 2.9 yields that R ≥ Λ. Combining this with R = Λ, we obtain R > Λ. Hence, (10) yields a R = 0. This contradicts a R = 0. This contradiction shows that our assumption was wrong. Hence, (11) is proven.] On the other hand, Lemma 2.46 (b) (applied to h = g and R = Λ) yields Therefore, This contradiction shows that our assumption was false. Hence, (a R ) R∈L n = (0) R∈L n is proven. Now, forget that we fixed (a R ) R∈L n . We thus have shown that if (a R ) R∈L n ∈ Q L n is a family of scalars (in Q) such that ∑ A second proof of Proposition 2.44 can be found in [Grinbe18].
Corollary 2.47. The family Proof of Corollary 2.47. Follows from Proposition 2.44 using gradedness; see [Grinbe18] for details.
We can now finally prove what we came here for: Theorem 2.48. The permutation statistic Epk is shuffle-compatible.
Question 2.50. Our concept of a "Z-enriched (P, γ)-partition" generalizes the concept of an "enriched (P, γ)-partition" by restricting ourselves to a subset Z of N × {+, −}. (This does not sound like much of a generalization when stated like this, but as we have seen the behavior of the power series Γ Z (P, γ) depends strongly on what Z is, and is not all anticipated by the Z = N × {+, −} case.) A different generalization of enriched (P, γ)partitions (introduced by Hsiao and Petersen in [HsiPet10]) are the colored (P, γ)-partitions, where the two-element set {+, −} is replaced by the set 1, ω, . . . , ω m−1 of all m-th roots of unity (where m is a chosen positive integer, and ω is a fixed primitive m-th root of unity). We can play various games with this concept. The most natural thing to do seems to be to consider m arbitrary total orders < 0 , < 1 , . . . , < m−1 on the codomain A of the labeling γ (perhaps with some nice properties such as all intervals being finite) and an arbitrary subset Z of N × 1, ω, . . . , ω m−1 , and define a Z-enriched colored (P, γ)-partition to be a map f : P → Z such that every x < y in P satisfy the following conditions: Is this a useful concept, and can it be used to study permutation statistics?
Question 2.51. Corollary 2.42 provides a formula for rewriting a product of the form K Z n,Λ · K Z m,Ω as a Q-linear combination of K Z n+m,Ξ 's when Λ ∈ L n and Ω ∈ L m (because any such Λ and Ω can be written as Λ = Epk π and Ω = Epk σ for appropriate permutations π and σ). Thus, in particular, any such product belongs to the Q-linear span of the K Z n+m,Ξ 's. Is this still true if Λ and Ω are arbitrary subsets of [n] and [m] rather than having to belong to L n and to L m ? Computations with SageMath suggest that the answer is "yes". For example, Note that the Q-linear span of the K Z n+m,Ξ 's for all Ξ ⊆ [n + m] is (generally) larger than that of the K Z n+m,Ξ 's with Ξ ∈ L n+m .

LR-shuffle-compatibility
In this section, we shall introduce the concept of "LR-shuffle-compatibility" (short for "left-and-right-shuffle-compatibility"), which is stronger than usual shuffle-compatibility. We shall prove that Epk still is LR-shuffle-compatible, and study some other statistics that are and some that are not.

Left and right shuffles
We begin by introducing "left shuffles" and "right shuffles". There is a wellknown notion of left and right shuffles of words (see, e.g., the operations ≺ and ≻ in [EbMaPa07, Example 1]). Specialized to permutations, it can be defined in the following simple way: Definition 3.1. Let π and σ be two disjoint permutations. Then: • A left shuffle of π and σ means a shuffle τ of π and σ such that the first letter of τ is the first letter of π. (This makes sense only when π is nonempty. Otherwise, there are no left shuffles of π and σ.) • A right shuffle of π and σ means a shuffle τ of π and σ such that the first letter of τ is the first letter of σ. (This makes sense only when σ is nonempty. Otherwise, there are no right shuffles of π and σ.) • We let S ≺ (π, σ) denote the set of all left shuffles of π and σ.
• We let S ≻ (π, σ) denote the set of all right shuffles of π and σ.
The permutations () and (1, 3) have only one right shuffle, which is (1, 3), and they have no left shuffles. Clearly, if π and σ are two disjoint permutations such that at least one of π and σ is nonempty, then the two sets S ≺ (π, σ) and S ≻ (π, σ) are disjoint and their union is S (π, σ) (because every shuffle of π and σ is either a left shuffle or a right shuffle, but not both).
Left and right shuffles have a recursive structure that makes them amenable to inductive arguments. To state it, we need one more definition: Definition 3.2. Let n ∈ N. Let π be an n-permutation.
Proof of Proposition 3.3. The fairly simple proof is left to the reader, who can also find it in [Grinbe18].

LR-shuffle-compatibility
We shall use the so-called Iverson bracket notation for truth values:

Definition 3.4. If A is any logical statement, then we define an integer [A] ∈ {0, 1} by
This integer [A] is known as the truth value of A.
We can now define a notion similar to shuffle-compatibility: Definition 3.5. Let st be a permutation statistic. We say that st is LR-shufflecompatible if and only if it has the following property: For any two disjoint nonempty permutations π and σ, the multisets depend only on st π, st σ, |π|, |σ| and [π 1 > σ 1 ].

Head-graft-compatibility
We shall now define another compatibility concept for a permutation statistic, which will later prove a useful stepping stone for checking the LR-shufflecompatibility of this statistic.
Definition 3.6. Let st be a permutation statistic. We say that st is head-graftcompatible if and only if it has the following property: For any nonempty permutation π and any letter a that does not appear in π, the element st (a : π) depends only on st π, |π| and [a > π 1 ].
(a) Let π be a nonempty permutation. Let a be a letter that does not appear in π. We shall express the element Des (a : π) in terms of Des π, |π| and [a > π 1 ].
• Adding 1 to each descent of π yields a descent of a : π. (That is, if i is a descent of π, then i + 1 is a descent of a : π.) These are all the descents of a : π. Thus, Des (a : π) = {1 | a > π 1 } ∪ (Des π + 1) .
(b) Let π be a nonempty permutation. Let a be a letter that does not appear in π. We shall express the element Lpk (a : π) in terms of Lpk π, |π| and [a > π 1 ].
Notice first that a = π 1 (since a does not appear in π). Thus, a < π 1 is true if and only if a > π 1 is false.
• Adding 1 to each left peak i of π yields a left peak i + 1 of a : π, except for the case when i = 1 (in which case i + 1 = 2 is a left peak of a : π only if a < π 1 ).
These are all the left peaks of a : π. Thus, This equality shows that Lpk (a : π) depends only on Lpk π, |π| and [a > π 1 ] (indeed, the truth value [a > π 1 ] determines whether a > π 1 is true and also determines whether a < π 1 is true 6 ). In other words, Lpk is head-graft-compatible (by the definition of "head-graft-compatible"). This proves Proposition 3.8 (b).
(c) To obtain a proof of Proposition 3.8 (c), it suffices to take our above proof of Proposition 3.8 (b) and replace every appearance of "left peak" and "Lpk" by "exterior peak" and "Epk".

Proving LR-shuffle-compatibility
Let us now state a sufficient criterion for the LR-shuffle-compatibility of a statistic: Theorem 3.9. Let st be a permutation statistic that is both shuffle-compatible and head-graft-compatible. Then, st is LR-shuffle-compatible.
Before we prove this theorem, let us introduce some terminology and state an almost-trivial fact:

Definition 3.10. (a) If
A is a finite multiset, and if g is any object, then |A| g means the multiplicity of g in A.
(b) If A and B are two finite multisets, then we say that B ⊆ A if and only if each object g satisfies |B| g ≤ |A| g .
(c) If A and B are two finite multisets satisfying B ⊆ A, then A − B shall denote the "multiset difference" of A and B; this is the finite multiset C such that each object g satisfies |C| g = |A| g − |B| g .
Proof of Theorem 3.9. We shall first show the following: Claim 1: Let π, π ′ and σ be three nonempty permutations. Assume that π and σ are disjoint. Assume that π ′ and σ are disjoint. Assume furthermore that Then, and [Proof of Claim 1: We shall prove Claim 1 by induction on |σ|: Induction base: The case |σ| = 0 cannot happen (because σ is assumed to be nonempty). Thus, Claim 1 is true in the case |σ| = 0. This completes the induction base.
Induction step: Let N be a positive integer. Assume (as the induction hypothesis) that Claim 1 holds when |σ| = N − 1. We must now prove that Claim 1 holds when |σ| = N.
Furthermore, Claim 2 (applied to γ instead of π) yields Finally, Claim 1 (applied to γ and σ ′ instead of π and σ) yields Combining the equalities we have found, we obtain The same argument (but with the symbols "S ≺ " and "S ≻ " interchanged) yields Thus, Claim 3 is proven in Case 1.

Some other statistics
The question of LR-shuffle-compatibility can be asked about any statistic. We have so far answered it for Des, Pk, Lpk, Rpk and Epk. In this section, we shall analyze it for some further statistics.

The descent number des
The permutation statistic des (called the descent number) is defined as follows: For each permutation π, we set des π = |Des π| (that is, des π is the number of all descents of π). It was proven in [GesZhu17, Theorem 4.6 (a)] that this statistic des is shuffle-compatible. We now claim the following: Proposition 3.13. The permutation statistic des is head-graft-compatible and LR-shuffle-compatible.
Proof of Proposition 3.13. From (12), we easily obtain the following: If π is a nonempty permutation, and if a is a letter that does not appear in π, then des (a : π) = des π + [a > π 1 ] .

The major index maj
The permutation statistic maj (called the major index) is defined as follows: For each permutation π, we set maj π = ∑ i∈Des π i (that is, maj π is the sum of all descents of π). It was proven in [GesZhu17, Theorem 3.1 (a)] that this statistic maj is shuffle-compatible.
3.5.3. The joint statistic (des, maj) The next permutation statistic we shall study is the so-called joint statistic (des, maj). This statistic is defined as the permutation statistic that sends each permutation π to the ordered pair (des π, maj π). (Calling it (des, maj) is thus a slight abuse of notation.) It was proven in [GesZhu17, Theorem 4.5 (a)] that this statistic (des, maj) is shuffle-compatible. We now claim the following: Proposition 3.14. The permutation statistic (des, maj) is head-graftcompatible and LR-shuffle-compatible.

The comajor index comaj
The permutation statistic comaj (called the comajor index) is defined as follows: For each permutation π, we set comaj π = ∑ k∈Des π (n − k), where n = |π|. It was proven in [GesZhu17, §3.2] that this statistic comaj is shuffle-compatible. We now claim the following: Proposition 3.15. The permutation statistic comaj is head-graft-compatible and LR-shuffle-compatible.

Left-and right-shuffle-compatibility
In this section, we shall study two notions closely related to LR-shuffle-compatibility: Definition 3.16. Let st be a permutation statistic.
For a shuffle-compatible permutation statistic, these two notions are equivalent to the notions of LR-shuffle-compatibility and head-graft-compatibility, as the following proposition reveals: Proposition 3.17. Let st be a shuffle-compatible permutation statistic. Then, the following assertions are equivalent: • Assertion A 1 : The statistic st is LR-shuffle-compatible.
• Assertion A 2 : The statistic st is left-shuffle-compatible.
• Assertion A 3 : The statistic st is right-shuffle-compatible.
• Assertion A 4 : The statistic st is head-graft-compatible.
Note that on their own, the properties of left-shuffle-compatibility and rightshuffle-compatibility are not equivalent. For example, the permutation statistic that sends each nonempty permutation π to the truth value [π 1 > π i for all i > 1] (and the 0-permutation () to 0) is right-shuffle-compatible (because in the definition of right-shuffle-compatibility, all the st τ will be 0), but not left-shufflecompatible.

Properties of compatible statistics
Let us state some more facts on compatibility properties. We refer to [Grinbe18] for their proofs. We begin with a converse to Theorem 3.9:

Descent statistics and quasisymmetric functions
In this section, we shall recall the concepts of descent statistics and their shuffle algebras (introduced in [GesZhu17]), and apply them to Epk.

Compositions
Definition 4.1. A composition is a finite list of positive integers. If I = (i 1 , i 2 , . . . , i n ) is a composition, then the nonnegative integer i 1 + i 2 + · · · + i n is called the size of I and is denoted by |I|; we furthermore say that I is a composition of |I|.
On the other hand, for each subset A = {a 1 < a 2 < · · · < a k } of [n − 1], we define a composition Comp A of n by Comp A = (a 1 , a 2 − a 1 , a 3 − a 2 , . . . , a k − a k−1 , n − a k ) .
(The definition of Comp A should be understood to give Comp A = (n) if A = ∅ and n > 0, and to give Comp A = () if A = ∅ and n = 0. Note that Comp A depends not only on the set A itself, but also on n. We hope that n will always be clear from the context when we use this notation. ) We thus have defined a map Des (from the set of all compositions of n to the set of all subsets of [n − 1]) and a map Comp (in the opposite direction). These two maps are mutually inverse bijections. Definition 4.3. Let n ∈ N. Let π = (π 1 , π 2 , . . . , π n ) be an n-permutation. The descent composition of π is defined to be the composition Comp (Des π) of n. This composition is denoted by Comp π.

Descent statistics
Definition 4.4. Let st be a permutation statistic. We say that st is a descent statistic if and only if st π (for π a permutation) depends only on the descent composition Comp π of π. In other words, st is a descent statistic if and only if every two permutations π and σ satisfying Comp π = Comp σ satisfy st π = st σ.
In [Oguz18, Corollary 1.6], Ezgi Kantarcı Oguz has demonstrated that not every shuffle-compatible permutation statistic is a descent statistic. However, this changes if we require LR-shuffle-compatibility, because of Corollary 3.18 and of the following fact: Proposition 4.5. Every head-graft-compatible permutation statistic is a descent statistic.
Definition 4.6. Let st be a descent statistic. Then, we can regard st as a map from the set of all compositions (rather than from the set of all permutations). Namely, for any composition I, we define st I (an element of the codomain of st) by setting st I = st π for any permutation π satisfying Comp π = I. This is well-defined (because for every composition I, there exists at least one permutation π satisfying Comp π = I, and all such permutations π have the same value of st π). In the following, we shall regard every descent statistic st simultaneously as a map from the set of all permutations and as a map from the set of all compositions.
Note that this definition leads to a new interpretation of Des I for a composition I: It is now defined as Des π for any permutation π satisfying Comp π = I. This could clash with the old meaning of Des I introduced in Definition 4.2. Fortunately, these two meanings of Des I are exactly the same, so there is no conflict of notation.
However, Definition 4.6 causes an ambiguity for expressions like "Des (i 1 , i 2 , . . . , i n )": Here, the "(i 1 , i 2 , . . . , i n )" might be understood either as a permutation, or as a composition, and the resulting descent sets Des (i 1 , i 2 , . . . , i n ) are not the same. A similar ambiguity occurs for any descent statistic st instead of Des. We hope that this ambiguity will not arise in this paper due to our explicit typecasting of permutations and compositions; but the reader should be warned that it can arise if one takes the notation too literally.

Quasisymmetric functions
We now recall the definition of quasisymmetric functions; see [GriRei18, Chapter 5] (and various other modern textbooks) for more details about this: • For any positive integers a 1 , a 2 , . . . , a k and any two strictly increasing sequences (i 1 < i 2 < · · · < i k ) and (j 1 < j 2 < · · · < j k ) of positive integers, the coefficient of x

(b)
A quasisymmetric function is a quasisymmetric power series f ∈ Q [[x 1 , x 2 , x 3 , . . .]] that has bounded degree (i.e., there exists an N ∈ N such that each monomial appearing in f has degree ≤ N).
(c) The quasisymmetric functions form a Q-subalgebra of Q [[x 1 , x 2 , x 3 , . . .]]; this Q-subalgebra is denoted by QSym and called the ring of quasisymmetric functions over Q. This Q-algebra QSym is graded (in the obvious way, i.e., by the degree of a monomial).
The Q-algebra QSym has much interesting structure (e.g., it is a Hopf algebra), some of which we will introduce later when we need it. One simple yet crucial feature of QSym that we will immediately use is the fundamental basis of QSym: Definition 4.9. For any composition α, we define the fundamental quasisymmetric function F α to be the power series ∑ i 1 ≤i 2 ≤···≤i n ; i j <i j+1 for each j∈Des α where n = |α| is the size of α. The family (F α ) α is a composition is a basis of the Q-vector space QSym; it is known as the fundamental basis of QSym.

Shuffle algebras
Any shuffle-compatible permutation statistic st gives rise to a shuffle algebra A st , defined as follows: Definition 4.11. Let st be a shuffle-compatible permutation statistic. For each permutation π, let [π] st denote the st-equivalence class of π.
Let A st be the free Q-vector space whose basis is the set of all st-equivalence classes of permutations. We define a multiplication on A st by setting [τ] st for any two disjoint permutations π and σ. It is easy to see that this multiplication is well-defined and associative, and turns A st into a Q-algebra whose unity is the st-equivalence class of the 0-permutation (). This Q-algebra is denoted by A st , and is called the shuffle algebra of st. It is a graded Q-algebra; its n-th graded component (for each n ∈ N) is spanned by the st-equivalence classes of all n-permutations.  (b) In this case, the Q-linear map where α is the st-equivalence class of the composition Comp π, is a Q-algebra isomorphism A st → A.
4.5. The shuffle algebra of Epk Theorem 2.48 yields that the permutation statistic Epk is shuffle-compatible. Hence, the shuffle algebra A Epk is well-defined. We have little to say about it: Proof of Theorem 4.14. See [Grinbe18].
We can describe A Epk using the notations of Section 2: Definition 4.15. Let Π Z be the Q-vector subspace of Pow N spanned by the family K Z n,Λ n∈N; Λ∈L n . Then, Π Z is also the Q-vector subspace of Pow N spanned by the family K Z n,Epk π n∈N; π is an n-permutation (by Proposition 2.5).
In other words, Π Z is also the Q-vector subspace of Pow N spanned by the family (Γ Z (π)) n∈N; π is an n-permutation (because of (8)). Hence, Corollary 2.29 shows that Π Z is closed under multiplication. Since furthermore Γ Z (()) = 1 (for the 0-permutation ()), we can thus conclude that Π Z is a Q-subalgebra of Pow N . [π] Epk → K Z n,Epk π is a Q-algebra isomorphism.

The kernel of a descent statistic
Now, we shall focus on a feature of shuffle-compatible descent statistics that seems to have been overlooked so far: their kernels. All proofs in this section are omitted; they can be found in [Grinbe18]. The following basic linear-algebraic lemma will be useful: Corollary 5.4. The kernel K Epk of the descent statistic Epk is an ideal of QSym.
We can study the kernel of any descent statistic; in particular, the case of shuffle-compatible descent statistics appears interesting. Since QSym is isomorphic to a polynomial ring (as an algebra), it has many ideals, which are rather hopeless to classify or tame; but the ones obtained as kernels of shufflecompatible descent statistics might be worth discussing.

An F-generating set of K Epk
Let us now focus on K Epk , the kernel of Epk.
Proposition 5.5. If J = (j 1 , j 2 , . . . , j m ) and K are two compositions, then we shall write J → K if there exists an ℓ ∈ {2, 3, . . . , m} such that j ℓ > 2 and K = (j 1 , j 2 , . . . , j ℓ−1 , 1, j ℓ − 1, j ℓ+1 , j ℓ+2 , . . . , j m ). (In other words, we write J → K if K can be obtained from J by "splitting" some entry j ℓ > 2 into two consecutive entries 7 1 and j ℓ − 1, provided that this entry was not the first entry -i.e., we had ℓ > 1 -and that this entry was greater than 2.) The ideal K Epk of QSym is spanned (as a Q-vector space) by all differences of the form F J − F K , where J and K are two compositions satisfying J → K.

Two operations on QSym
We begin with some definitions. We will use some notations from [Grinbe16], but we set k = Q because we are working over the ring Q in this paper. Monomials always mean formal expressions of the form x a 1 1 x a 2 2 x a 3 3 · · · with a 1 + a 2 + a 3 + · · · < ∞ (see [Grinbe16, Section 2] for details). If m is a monomial, then Supp m will denote the finite subset {i ∈ {1, 2, 3, . . .} | the exponent with which x i occurs in m is > 0} of {1, 2, 3, . . .}. Next, we define two binary operations ≺ (called "dendriform less-than"; but it's an operation, not a relation) , (called "dendriform greater-or-equal"; but it's an operation, not a relation) , (a ≺ b) ≺ c = a ≺ (bc) ; (a b) ≺ c = a (b ≺ c) ; a (b c) = (ab) c.
The operations ≺ and are sometimes called "restricted products" due to their similarity with the (regular) multiplication of QSym. In particular, they satisfy the following analogue of Proposition 4.10: Proposition 6.1. Let π and σ be two disjoint nonempty permutations. Assume that π 1 > σ 1 . Then,

Left-and right-shuffle-compatibility and ideals
This proposition lets us relate the notions introduced in Definition 3.16 to the operations ≺ and . To state the precise connection, we need the following notation: Definition 6.2. Let A be a k-module equipped with some binary operation * (written infix). (a) We say that st is weakly left-shuffle-compatible if for any two disjoint nonempty permutations π and σ having the property that each entry of π is greater than each entry of σ, the multiset {st τ | τ ∈ S ≺ (π, σ)} multi depends only on st π, st σ, |π| and |σ|.
(b) We say that st is weakly right-shuffle-compatible if for any two disjoint nonempty permutations π and σ having the property that each entry of π is greater than each entry of σ, the multiset {st τ | τ ∈ S ≻ (π, σ)} multi depends only on st π, st σ, |π| and |σ|.
Then, the following analogues to the first part of Proposition 5.3 hold: Theorem 6.4. Let st be a descent statistic. Then, the following three statements are equivalent: • Statement A: The statistic st is left-shuffle-compatible.
• Statement B: The statistic st is weakly left-shuffle-compatible.
• Statement C: The set K st is an ≺ -ideal of QSym.
Theorem 6.5. Let st be a descent statistic. Then, the following three statements are equivalent: • Statement A: The statistic st is right-shuffle-compatible.
• Statement B: The statistic st is weakly right-shuffle-compatible.
• Statement C: The set K st is an -ideal of QSym.
Corollary 6.6. Let st be a permutation statistic that is LR-shuffle-compatible. Then, st is a shuffle-compatible descent statistic, and the set K st is an ideal and a ≺ -ideal and a -ideal of QSym.
Corollary 6.7. Let st be a descent statistic such that K st is a ≺ -ideal and a -ideal of QSym. Then, st is LR-shuffle-compatible and shuffle-compatible.
Corollary 6.6 can (for example) be applied to st = Epk, which we know to be LR-shuffle-compatible (from Theorem 3.12 (c)); the result is that K Epk is an ideal and a ≺ -ideal and a -ideal of QSym. The same can be said about Des and Lpk and some other statistics.
Combining Theorem 6.4 with Theorem 6.5, we can also see that any descent statistic that is weakly left-shuffle-compatible and weakly right-shufflecompatible must automatically be shuffle-compatible (because any ≺ -ideal of QSym that is also a -ideal of QSym is an ideal of QSym as well). Note that this is only true for descent statistics! As far as arbitrary permutation statistics are concerned, this is false; for example, the number of inversions is weakly left-shuffle-compatible and weakly right-shuffle-compatible but not shuffle-compatible.
Let us next define the notion of dendriform algebras: Definition 6.8. (a) A dendriform algebra over a field k means a k-algebra A equipped with two further k-bilinear binary operations ≺ and (these are operations, not relations, despite the symbols) from A × A to A that satisfy the four rules a ≺ b + a b = ab; (a ≺ b) ≺ c = a ≺ (bc) ; (a b) ≺ c = a (b ≺ c) ; a (b c) = (ab) c Question 6.11. Can the Q-algebra Pow N from Definition 2.19 be endowed with two binary operations ≺ and that make it into a dendriform algebra? Can we then find an analogue of Proposition 2.24 along the following lines? Let (P, γ), (Q, δ) and (P ⊔ Q, ε) be as in Proposition 2.24. Assume that each of the posets P and Q has a (global) minimum element; denote these elements by min P and min Q, respectively. Let P ≺ Q be the poset obtained by adding the relation min P < min Q to P ⊔ Q. Let P ≻ Q be the poset obtained by adding the relation min P > min Q to P ⊔ Q. Then, we hope to have Γ Z (P, γ) ≺ Γ Z (Q, δ) = Γ Z (P ≺ Q, ε) and Γ Z (P, γ) Γ Z (Q, δ) = Γ Z (P ≻ Q, ε) , assuming a simple condition on min P and min Q (say, γ (min P) < Z δ (min Q)).