Cyclic shuffle-compatibility via cyclic shuffle algebras

A permutation statistic $\operatorname{st}$ is said to be shuffle-compatible if the distribution of $\operatorname{st}$ over the set of shuffles of two disjoint permutations $\pi$ and $\sigma$ depends only on $\operatorname{st}\pi$, $\operatorname{st}\sigma$, and the lengths of $\pi$ and $\sigma$. Shuffle-compatibility is implicit in Stanley's early work on $P$-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.


Introduction
We say that π = π 1 π 2 • • • π n is a (linear ) permutation of length n if it is a sequence of n distinct letters-not necessarily from 1 to n-in P, the set of positive integers.(We refer to these as linear permutations to distinguish them from cyclic permutations, but we will often drop the descriptor "linear" if it is clear from context that we are referring to linear permutations.)For example, 826491 is a permutation of length 6.Let |π| denote the length of a permutation π, let P n denote the set of permutations of length n, and S n ⊆ P n the set of permutations of [n] := {1, 2, . . ., n}.Note that P 0 and S 0 consist only of the empty word.
Let π ∈ P m and σ ∈ P n be disjoint permutations, that is, permutations with no letters in common.We say that τ ∈ P m+n is a shuffle of π and σ if both π and σ are subsequences of τ .The set of shuffles of π and σ is denoted π ¡ σ.For example, 71 ¡ 25 = {7125, 7215, 7251, 2715, 2751, 2571}.
Following [GZ18], a (linear ) permutation statistic is a function st on permutations such that st π = st σ whenever π and σ have the same relative order.1 Three classical permutation statistics, dating back to MacMahon [Mac60], are the descent set Des, descent number des, and the major index maj.We say that i ∈ is the set of its descents, the descent number des π := |Des π| its number of descents, and the major index maj π := i∈Des π i the sum of its descents.
Several other permutation statistics-somewhat less classical but still well-studied-are based on the notion of peaks.We say that i ∈ {2, 3, . . ., n − 1} is a peak of π ∈ P n if π i−1 < π i > π i+1 .The peak set of π Pk π := { i ∈ {2, 3, . . ., n − 1} : π i−1 < π i > π i+1 } is the set of its peaks, and the peak number pk π := |Pk π| is its number of peaks.Some related statistics, such as the left peak set and left peak number, will be defined in Section 5.4.
Given a set S of permutations and a permutation statistic st, the distribution of st over S is the multiset st S := {{ st π : π ∈ S }} of all values of st among permutations in S, including multiplicity.For instance, des S 3 = {{0, 1 4 , 2}}; among the six permutations in S 3 , only 123 has no descents, only 321 has two descents, and the other four have one descent each.
Shuffle-compatibility dates back to the early work of Stanley, as the shuffle-compatibility of the descent set, descent number, and major index are implicit consequences of the theory of P -partitions [Sta72].Likewise, Stembridge's work on enriched P -partitions imply that the peak set and peak number are shuffle-compatible.Gessel and Zhuang coined the term "shuffle-compatibility" and initiated the study of shuffle-compatibility per se in 2018; in [GZ18], they developed an algebraic framework for shuffle-compatibility centered around the notion of the shuffle algebra of a shuffle-compatible permutation statistic, which is welldefined if and only if the statistic is shuffle-compatible and whose multiplication encodes the distribution of the statistic over sets of shuffles.
Gessel's [Ges84] quasisymmetric functions serve as natural generating functions for Ppartitions, and for a special family of statistics called "descent statistics", one can use quasisymmetric functions to characterize shuffle algebras and prove shuffle-compatibility results.Notably, the multiplication rule for fundamental quasisymmetric functions shows that the descent set is shuffle-compatible and that its shuffle algebra is isomorphic to the algebra QSym of quasisymmetric functions.One of Gessel and Zhuang's main results is a necessary and sufficient condition for shuffle-compatibility of descent statistics which implies that the shuffle algebra of any shuffle-compatible descent statistic is isomorphic to a quotient algebra of QSym.
In analogy to linear permutation statistics, let us define a cyclic permutation statistic to be a function cst on cyclic permutations such that cst[π] = cst[σ] whenever π and σ have the same relative order.Two examples of cyclic permutation statistics are the cyclic descent set cDes and the cyclic descent number cdes.First, define the cyclic descent set of a linear permutation π ∈ P n by cDes π := { i ∈ [n] : π i > π i+1 where i is considered modulo n }; the elements of cDes π are called cyclic descents of π.The cyclic descent set of a cyclic permutation [π] is the multiset cDes[π] := {{ cDes π : π ∈ [π] }}, i.e., the distribution of the linear statistic cDes over all linear permutation representatives of [π].For example, we have cDes [168425] = {{ {3, 4, 6}, {1, 4, 5}, {2, 5, 6}, {1, 3, 6}, {1, 2, 4}, {2, 3, 5} }}, and cDes[279358] = {{ {3, 6} 2 , {1, 4} 2 , {2, 5} 2 }.Note that cDes[π] can also be characterized as the multiset of cyclic shifts of cDes π.More precisely, given S ⊆ [n] and an integer i, define the cyclic shift S + i by where the values are considered modulo n; then The cyclic descent number of a linear permutation π is given by cdes π := |cDes π| , and we can then define the cyclic descent number of a cyclic permutation [π] by cdes[π] := cdes π, which is well-defined because all linear permutations in [π] have the same number of cyclic descents.The cyclic peak set cPk and cyclic peak number cpk can be defined in an analogous way, and we will state their definitions in Section 4.1.On the other hand, finding a suitable cyclic analogue of the major index statistic is challenging; we will address this in Section 5.3.
The first results in cyclic shuffle-compatibility were implicit in the work of Adin et al. [AGRR21], which introduced toric [ D]-partitions (a toric poset analogue of P -partitions) and cyclic quasisymmetric functions (which are natural generating functions for toric [ D]partitions).In particular, Adin et al. established a multiplication formula for fundamental cyclic quasisymmetric functions which implies that the cyclic descent set cDes is cyclic shufflecompatible, and they also proved the formula which implies that the cyclic descent number cdes is cyclic shuffle-compatible.In [DLM + 21], Domagalski et al. formally defined cyclic shuffle-compatibility and proved a result called the "lifting lemma," which allows one (under certain nice conditions) to prove that a cyclic statistic is cyclic shuffle-compatible from the shuffle-compatibility of a related linear statistic.They then used the lifting lemma to prove the cyclic shuffle-compatibility of all four statistics cDes, cdes, cPk, and cpk.
Most recently, Liang [Lia22] defined and studied enriched toric [ D]-partitions, an analogue of enriched P -partitions for toric posets, whose generating functions are "cyclic peak quasisymmetric functions".She derived a multiplication formula for these cyclic peak quasisymmetric functions which gives a different proof for the cyclic shuffle-compatibility of the cyclic peak set cPk.
The lifting lemma of Domagalski et al. is purely combinatorial, but the work of Adin et al. and Liang suggest that there is an algebraic framework for cyclic shuffle-compatibility à la Gessel and Zhuang, in which the role of quasisymmetric functions is replaced by cyclic quasisymmetric functions.The goal of our paper is to develop this algebraic framework.
See [LSZ23] for an extended abstract of this work.
1.2.Outline.The organization of this paper is as follows.In Section 2, we review Gessel and Zhuang's definition of the shuffle algebra of a shuffle-compatible permutation statistic, and then we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic.We prove several general results about cyclic shuffle-compatibility via cyclic shuffle algebras, including a result (Theorem 2.8) allowing one to construct cyclic shuffle algebras from linear ones.
In Section 3, we review the role of quasisymmetric functions in the theory of (linear) shufflecompatibility, and then we develop an analogous theory concerning cyclic quasisymmetric functions and cyclic shuffle-compatibility.We use Theorem 2.8 to construct the non-Escher subalgebra cQSym − of cyclic quasisymmetric functions from the algebra QSym of quasisymmetric functions, which gives another proof that cDes is cyclic shuffle-compatible and shows that the cyclic shuffle algebra of cDes is isomorphic to cQSym − .We then give a necessary and sufficient condition for cyclic shuffle-compatibility of cyclic descent statistics which implies that the cyclic shuffle algebra of any cyclic shuffle-compatible cyclic descent statistic is isomorphic to a quotient algebra of cQSym − .
In Section 4, we use the theory developed in Section 3 to give explicit descriptions of the shuffle algebras of the statistics cPk, cpk cdes, and (cpk, cdes) which in turn yields algebraic proofs for their cyclic shuffle-compatibility.
In Section 5, we define a family of multiset-valued cyclic statistics induced from linear statistics, and investigate cyclic shuffle-compatibility for some of these statistics.This approach yields a definition of a cyclic major index which is different from the one proposed earlier by Ji and Zhang [JZ22]; unfortunately, neither of these cyclic major index statistics are cyclic shuffle-compatible.
We conclude the paper in Section 6 with a discussion of open problems and questions related to our work.

Cyclic shuffle algebras
At the heart of Gessel and Zhuang's algebraic framework for shuffle-compatibility is the notion of a shuffle algebra.In this section, we review the definition of the shuffle algebra of a shuffle-compatible (linear) permutation statistic, define a cyclic analogue of shuffle algebras for cyclic shuffle-compatible statistics, and prove several general results about cyclic shufflecompatibility through cyclic shuffle algebras, including one that can be used to construct cyclic shuffle algebras from shuffle algebras of linear permutation statistics.
2.1.Definitions.Let st be a permutation statistic.We say that π and σ are st-equivalent if st π = st σ and |π| = |σ|.In this way, every permutation statistic induces an equivalence relation on permutations, and we write the st-equivalence class of π as π st . 2et A st denote the Q-vector space consisting of formal linear combinations of st-equivalence classes of permutations.If st is shuffle-compatible, then we can turn A st into a Q-algebra by endowing it with the multiplication for any disjoint representatives π ∈ π st and σ ∈ σ st ; this multiplication is well-defined (i.e., the choice of π and σ does not matter) precisely when st is shuffle-compatible.The Q-algebra A st is called the (linear ) shuffle algebra of st.Observe that A st is graded by length, that is, π st belongs to the nth homogeneous component of A st if π has length n.
Our definition of cyclic shuffle algebras will be analogous to that of linear ones.Let cst be a cyclic permutation statistic.Then the cyclic permutations [π] and [σ] are called cstequivalent if cst[π] = cst[σ] and |π| = |σ|, and we use the notation [π] cst to denote the cst-equivalence class of the cyclic permutation [π].We associate to cst a Q-vector space A cyc cst by taking as a basis the set of all cst-equivalence classes of permutations, and then we give this vector space a multiplication by defining [τ ] cst for any disjoint π and σ with [π] ∈ [π] cst and [σ] ∈ [σ] cst ; this multiplication is well-defined if and only if cst is cyclic shuffle-compatible.The resulting Q-algebra A cyc cst is called the cyclic shuffle algebra of cst, and is also graded by length.
2.2.Two general results on cyclic shuffle algebras.We now give two general results on cyclic shuffle algebras, which are analogous to Theorems 3.2 and 3.3 of [GZ18] on linear shuffle algebras.We provide proofs for completeness, although they follow in essentially the same way as the proofs of the corresponding results in [GZ18].
Given two cyclic permutation statistics cst 1 and cst 2 , we say that cst 1 is a refinement of cst 2 if for all cyclic permutations [π] and [σ] of the same length, cst ; when this is true, we also say that cst 2 is a coarsening of cst 1 .Coarsenings of the cyclic descent set are called cyclic descent statistics.
Theorem 2.1.Suppose that cst 1 is cyclic shuffle-compatible and is a refinement of cst 2 .Let A be a Q-algebra with basis {v α } indexed by cst 2 -equivalence classes α, and suppose that there exists a Q-algebra homomorphism φ : A cyc cst 1 → A such that for every cst 1 -equivalence class β, we have φ(β) = v α where α is the cst 2 -equivalence class containing β. Then cst 2 is cyclic shuffle-compatible and the map v α → α extends by linearity to an isomorphism from A to A cyc cst 2 .
Proof.It suffices to show that for any disjoint π and σ, we have To that end, we have which completes the proof.
We say that cst 1 and cst 2 are equivalent if cst 1 is a simultaneously a refinement and a coarsening of cst 2 , that is, if for all cyclic permutations [π] and [σ] of the same length, cst Theorem 2.2.Let cst 1 and cst 2 be equivalent cyclic permutation statistics.If cst 1 is cyclic shuffle-compatible with cyclic shuffle algebra A cyc cst 1 , then cst 2 is also cyclic shuffle-compatible with cyclic shuffle algebra A cyc cst 2 isomorphic to A cyc cst 1 .
Proof.Because equivalent statistics have the same equivalence classes on cyclic permutations, we know that A cyc cst 1 and A cyc cst 2 have the same basis elements.Since cst 1 and cst 2 are equivalent, we have [τ ] st 2 , which proves the result.2.3.Symmetries and cyclic shuffle algebras.Many permutation statistics-both linear and cyclic-are related via various symmetries, such as reversal, complementation, and reverse-complementation.For a linear permutation π = π 1 π 2 • • • π n ∈ P n , we define the reversal π r of π by π r := π n π n−1 • • • π 1 , the complement π c of π to be the permutation obtained by (simultaneously) replacing the ith smallest letter in π with the ith largest letter in π for all 1 ≤ i ≤ n, and the reverse-complement π rc of π by π rc := (π r ) c = (π c ) r .For example, given π = 318269, we have π r = 962813, π c = 692831, and π rc = 138296.
More generally, let f be an involution on linear permutations which preserves the length, i.e., |f (π)| = |π| for all π.We shall write π f in place of f (π).For a set S of permutations, let S f := { π f : π ∈ S }, so f induces an involution on sets of permutations as well.In particular, this lets us define [π] f for a cyclic permutation [π].Going further, if C is a set of cyclic permutations, then Following Gessel and Zhuang [GZ18], we say that f is shuffle-compatibility-preserving if for any pair of disjoint permutations π and σ, there exist disjoint permutations π and σ with the same relative order as π and σ, respectively, such that (π ¡ σ) f = πf ¡ σf and (π ¡ σ) f = π f ¡ σ f .(This definition implies that π f and σ f are disjoint, and similarly with πf and σf .)Furthermore, we call two linear permutation statistics st 1 and st 2 f -equivalent if st 1 •f is equivalent to st 2 -that is, st 1 π f = st 1 σ f if and only if st 2 π = st 2 σ.In other words, st 1 and st 2 are f -equivalent if and only if (π f ) st 1 = (π st 2 ) f for all π.It is easy to see that, if st 1 π f = st 2 π for all π, then st 1 and st 2 are f -equivalent (although this is not a necessary condition).
For example, the peak set Pk is c-equivalent to the valley set Val defined in the following way.We call i ∈ {2, 3, . . ., n − 1} a valley of π ∈ P n if π i−1 > π i < π i+1 , and we let Val π be the set of valleys of π.We also define val π to be the number of valleys of π; then, pk and val are c-equivalent as well.
Despite its name, f -equivalence is not an equivalence relation (although it is symmetric).However, it turns out that if the statistics involved are shuffle-compatible, then fequivalences induce isomorphisms on the corresponding shuffle algebras.This idea is expressed in the following theorem, which is Theorem 3.5 of Gessel and Zhuang [GZ18].
Theorem 2.3.Let f be shuffle-compatibility-preserving, and suppose that st 1 and st 2 are f -equivalent (linear ) permutation statistics.If st 1 is shuffle-compatible with shuffle algebra A st 1 , then st 2 is also shuffle-compatible, and the linear map defined by π st 1 → π f st 2 is a Q-algebra isomorphism between their shuffle algebras A st 1 and A st 2 .
Gessel and Zhuang proved that reversal, complementation, and reverse-complementation are all shuffle-compatibility-preserving.Thus, they were able to use Theorem 2.3 to prove a collection of shuffle-compatibility results for statistics that are r-, c-, or rc-equivalent to another statistic whose shuffle-compatibility had already been established.For example, it follows from the shuffle-compatibility of the peak set Pk that the valley set Val is shufflecompatible with shuffle algebra A Val isomorphic to A Pk .
Moving onto the cyclic setting, let us call f rotation-preserving if [π] f = [π f ] for all π.We now prove that if f is both shuffle-compatibility-preserving and rotation-preserving, then f satisfies a cyclic version of the shuffle-compatibility-preserving property.
Lemma 2.4.If f is shuffle-compatibility-preserving and rotation-preserving, then for any pair of disjoint permutations π and σ, there exist disjoint permutations π and σ with the same relative order as π and σ, respectively, for which , so that τ ∈ π ¡ σ for some π ∈ [π] and σ ∈ [σ], and thus τ f ∈ (π ¡ σ) f .Since f is shuffle-compatibility-preserving, we have that τ f ∈ πf ¡ σf where π and σ are disjoint permutations with the same relative order as π and σ, respectively.Since π is a rotation of π and π has the same relative order as π, it follows that π is a rotation of a permutation π with the same relative order as π, and similarly σ is a rotation of a permutation σ with the same relative order as σ.Clearly, π and σ are disjoint because π and σ are disjoint.Because We have shown that but since these two sets have the same cardinality, they are in fact equal.We omit the proof of ( as it is similar.
so reversal is rotation-preserving.Moreover, it is clear that taking the complement of the permutation obtained by rotating the last n − i letters of π to the front) yields the same result as first taking the complement of π and then rotating the last n − i letters of π c to the front, so complementation is rotation-preserving.Lastly, since we have established that [π c ] = [π] c for all permutations π, we can replace π by π r to obtain [π rc ] = [π r ] c = [π] rc , so reverse-complementation is rotation-preserving as well.
In analogy with f -equivalence of linear permutation statistics, let us call two cyclic permutation statistics cst 1 and cst Theorem 2.6.Let f be shuffle-compatibility-preserving and rotation-preserving, and let cst 1 and cst 2 be f -equivalent cyclic permutation statistics.If cst 1 is cyclic shuffle-compatible, then cst 2 is cyclic shuffle-compatible with A cyc cst 2 isomorphic to A cyc cst 1 .Proof.Let [π] and [π] be cyclic permutations in the same cst 2 -equivalence class, and similarly with [σ] and [σ], such that π and σ are disjoint and π and σ are disjoint.We know from Lemma 2.4 that there exist permutations π, σ, π, and σ-having the same relative order as π, σ, π, and σ, respectively-satisfying ( Because π and π have the same relative order as π and π, respectively, we have Then, because cst 1 and cst 2 are f -equivalent, we have and [ πf ] are cst 1 -equivalent.The same reasoning shows that [σ f ] and [ σf ] are also cst 1 -equivalent.
By cyclic shuffle-compatibility of cst 1 , we have the multiset equality which-by f -equivalence of cst 1 and cst 2 -is equivalent to which is in turn equivalent to we have [τ ] cst 2 because cst 2 is cyclic shuffle-compatible, and thus we have Hence, λ is a Q-algebra isomorphism from A cyc cst 2 to A cyc cst 1 .Corollary 2.7.Suppose that the cyclic permutation statistics cst 1 and cst 2 are r-equivalent, c-equivalent, or rc-equivalent.If cst 1 is cyclic shuffle-compatible, then cst 2 is cyclic shufflecompatible with cyclic shuffle algebra A cyc cst 2 isomorphic to A cyc cst 1 .

Constructing cyclic shuffle algebras from linear ones.
The following theoremone of the main results of this paper-allows us to construct cyclic shuffle algebras from shuffle algebras of shuffle-compatible (linear) permutation statistics.
Theorem 2.8.Let cst be a cyclic permutation statistic and let st be a shuffle-compatible (linear ) permutation statistic.Given a cyclic permutation [π], let and [σ] are cst-equivalent, and that {v [π] } (ranging over all cst-equivalence classes) is linearly independent.Then cst is cyclic shuffle-compatible and the map ψ cst : extends linearly to a Q-algebra isomorphism from A cyc cst to the span of {v whenever [π] and [σ] are cst-equivalent, we know that ψ cst is a welldefined linear map on A cyc cst .(We do not yet know whether A cyc cst is an algebra; here we are only considering A cyc cst as a vector space.)Furthermore, because {v [π] } is linearly independent, the linear map ψ cst is a vector space isomorphism from A cyc cst to a subspace of A st .To show that cst is cyclic shuffle-compatible, we show that , where π ′ and σ ′ are disjoint and so are π ′′ and σ ′′ .Then and similarly Since [π ′ ] and [π ′′ ] are cst-equivalent and similarly with [σ ′ ] and [σ ′′ ], we have [τ ] cst due to injectivity of ψ cst .We have shown that the multiplication of the cyclic shuffle algebra A cyc cst is well-defined, and therefore cst is shuffle-compatible.Finally, we have

Shuffle-compatibility and quasisymmetric functions
The focus of this section is the relationship between cyclic shuffle-compatibility and cyclic quasisymmetric functions.We shall begin by providing the necessary background on descent compositions, cyclic descent compositions, and (ordinary) quasisymmetric functions.
3.1.Descent compositions.Every permutation can be uniquely decomposed into a sequence of maximal increasing consecutive subsequences, which we call increasing runs.For example, the increasing runs of 4783291 are 478, 3, 29, and 1. Equivalently, an increasing run of π is a maximal consecutive subsequence with no descents.
The number of increasing runs of a nonempty permutation is one more than its number of descents; in fact, the lengths of the increasing runs determine the descents, and vice versa.Given a subset S ⊆ [n − 1] with elements s 1 < s 2 < • • • < s j , let Comp S be the composition of n defined by Comp S := (s 1 , s 2 − s 1 , . . ., s j − s j−1 , n − s j ); also, given a composition L = (L 1 , L 2 , . . ., L k ), let It is straightforward to verify that Comp and Des are inverse bijections.If π ∈ P n has descent set S ⊆ [n − 1], then we say that Comp S is the descent composition of π, which we also denote by Comp π.By convention, the empty permutation has descent composition ∅.
Continuing the example above, we have Comp 4783291 = (3, 1, 2, 1).Observe that the descent composition of π gives the lengths of the increasing runs of π in the order that they appear.Conversely, if π has descent composition L, then its descent set Des π is Des L.
We call a permutation statistic st a descent statistic if it depends only on the descent composition, that is, if Comp π = Comp σ implies st π = st σ.Equivalently, a descent statistic depends only on the descent set and length.If st is a descent statistic, then we can extend the notion of st-equivalence classes of permutations to that of compositions.First, let st L indicate the value of st on any permutation with descent composition L. Then we say that two compositions L and K of the same size-where the size of a composition is the sum of its parts-are st-equivalent if st L = st K.For example, the compositions (2, 3, 1) and (1, 1, 4) are des-equivalent because any permutation with one of these descent compositions has exactly two descents.
3.2.Cyclic descent compositions.The notion of descent compositions for linear permutations can be extended to cyclic permutations.To do so, we shall need a few more preliminary definitions.A cyclic shift of a composition L = (L 1 , L 2 , . . ., L k ) is a composition of the form (L j , L j+1 , . . ., L k , L 1 , . . ., L j−1 ).
Let us call S a non-Escher3 subset of [n] if S is the cyclic descent set of some linear permutation of length n.When n = 0 or n = 1, only the empty set is non-Escher, and when n ≥ 2, all subsets of [n] are non-Escher except for the empty set and [n] itself.We associate to each non-Escher subset S ⊆ [n] a composition cComp S defined by A cyclic permutation statistic cst is called a cyclic descent statistic if it depends only on the cyclic descent composition-that is, if cComp (This is equivalent to the definition given in Section 2.2.)Similar to the notation st L, we can write cst [L] for the value of cst on any cyclic permutation with cyclic descent composition [L], and we shall say that two cyclic compositions [L] and [K] of the same size-which means that L and K have the same size-are cst-equivalent if cst The Q-vector space QSym n of quasisymmetric functions homogeneous of degree n has dimension 2 n−1 , the number of compositions of n.An important basis of QSym n is the basis of fundamental quasisymmetric functions {F n,L } L n defined by Sometimes, it is more convenient to index fundamental quasisymmetric functions by subsets of [n − 1] as opposed to compositions of n, in which case we'll use the notation The product of two quasisymmetric functions is again quasisymmetric.The multiplication rule for the fundamental basis is given by the following theorem, which can be proved using P -partitions; see [Sta01, Exercise 7.93].
Theorem 3.1.Let m and n be non-negative integers, and let where π is any permutation of length m with descent set A and σ is any permutation (disjoint from π) of length n with descent set B.
Motivated by Stanley's theory of P -partitions, Gessel introduced quasisymmetric functions in [Ges84] and developed the basic algebraic properties of QSym.Further properties of QSym and its connections with many topics of study in combinatorics and algebra were developed in the subsequent decades; see [GR20, Section 5], [LMvW13], [Sag20, Chapter 8], and [Sta11, Section 7.19] for several basic references.
From Theorem 3.1, we see that the descent set shuffle algebra A Des is isomorphic to QSym; this is Corollary 4.2 of [GZ18].
3.4.Cyclic quasisymmetric functions and the cyclic shuffle algebra of cDes.We are now ready to discuss cyclic quasisymmetric functions and their role in cyclic shufflecompatibility.
Given a subset S of [n] where n ≥ 1, let and let F cyc 0,∅ := 1; these are the fundamental cyclic quasisymmetric functions introduced by Adin, Gessel, Reiner, and Roichman [AGRR21].It is clear from this definition that the F cyc n,S are invariant under cyclic shift; in other words, if S ′ = S + i for some integer i, then is the equivalence class of the set S under cyclic shift, then it makes sense to define F cyc n,[S] := F cyc n,S .We can also index fundamental cyclic quasisymmetric functions using compositions; for a composition L of n, let Note that n is not needed in the subscript when using L or [L] since it is determined from the sum of the parts of L.

Let cQSym
where Proof.We know that the descent set Des is shuffle-compatible and its shuffle algebra A Des is isomorphic to the algebra of quasisymmetric functions, QSym, through the isomorphism φ Des (π Des ) = F |π|,Des(π) .Then, using the notation of Theorem 2.8, we have , denoted cQSym, is a graded Q-subalgebra of QSym, although this result is less relevant to cyclic shuffle-compatibility.Thus we have the subalgebra relations cQSym − ⊆ cQSym ⊆ QSym, and cQSym − is called the non-Escher subalgebra of cQSym.
Before moving on, let us explicitly state the cyclic shuffle-compatibility of cDes as a corollary of the preceding theorem.
Corollary 3.3 (Cyclic shuffle-compatibility of cDes).The cyclic descent set cDes is cyclic shuffle-compatible, and the linear map on A cyc cDes defined by

A general cyclic shuffle-compatibility criterion for cyclic descent statistics.
The theorem below is [GZ18, Theorem 4.3], which provides a necessary and sufficient condition for shuffle-compatibility of descent statistics in terms of quasisymmetric functions, and implies that the shuffle algebra of any shuffle-compatible descent statistic is a quotient algebra of QSym.
Theorem 3.4.A descent statistic st is shuffle-compatible if and only if there exists a Qalgebra homomorphism φ st : QSym → A, where A is a Q-algebra with basis {u α } indexed by st-equivalence classes α of compositions, such that φ st (F L ) = u α whenever L ∈ α.In this case, the linear map on A st defined by where Comp π ∈ α, is a Q-algebra isomorphism from A st to A.
We now prove our main result of this section: a cyclic analogue of Theorem 3.4.
Theorem 3.5.A cyclic descent statistic cst is cyclic shuffle-compatible if and only if there exists a Q-algebra homomorphism φ cst : cQSym − → A, where A is a Q-algebra with basis {v α } indexed by cst-equivalence classes α of non-Escher cyclic compositions, such that In this case, the linear map on A cyc cst defined by Suppose that the cyclic descent statistic cst is cyclic shuffle-compatible.Let A = A cyc cst be the cyclic shuffle algebra of cst, and let where c α β,γ is the number of cyclic permutations with cyclic descent composition in α that are obtained as a cyclic shuffle of two disjoint cyclic permutations, one with cyclic descent composition in β and the other with cyclic descent composition in γ.Observe that c α β,γ = [L]∈α c L J,K for any choice of [J] ∈ β and [K] ∈ γ, where c L J,K is the number of cyclic permutations with cyclic descent composition [L] that are obtained as a cyclic shuffle of two disjoint cyclic permutations, one with cyclic descent composition [J] and the other with cyclic descent composition [K].
Define the linear map which is well-defined because every linear permutation in [π] has the same number of cyclic peaks.It is easy to see that cPk and cpk are both cyclic descent statistics, so they are uniquely determined by the cyclic descent composition (equivalently, the cyclic descent set and length).When we characterize the (cpk, cdes) cyclic shuffle algebra, we shall need to determine all values that the (cpk, cdes) statistic can take, which we can do with the help of two lemmas.The first of these lemmas is Proposition 2.5 of [GZ18], so we omit its proof.Proof.Every peak of π is a cyclic peak of πm, and every cyclic peak of πm is either m or a peak of π.The same relationship is true for descents of π and cyclic descents of πm.Then part (a) follows from these equations and Lemma 4.1 (a).
To prove part (b), let j and k be integers in the specified ranges.By Lemma 4.1 (b), we know there exists a permutation π ′ ∈ P n−1 with pk π ′ = j − 1 and des π ′ = k − 1.Let m ∈ P be greater than the largest letter of π ′ ; then it follows from Lemma 4.2 that πm is a permutation in P n satisfying cpk π = j and cdes π = k.

4.2.
The cyclic shuffle algebra of cPk.We will construct the cyclic shuffle algebra A cyc cPk from the linear shuffle algebra A Pk .The latter is known to be isomorphic to a subalgebra Π of QSym-introduced by Stembridge [Ste97]-called the algebra of peaks, which is spanned by the peak quasisymmetric functions K n,S where n ranges over all non-negative integers and S over all possible peak sets of permutations in P n .We won't need the precise definition of K n,S here, only that the isomorphism from A Pk to Π sends π Pk to K |π|,Pk π .We state this fact in the following theorem, which appears as Theorem 4.7 of [GZ18].
Theorem 4.4 (Shuffle-compatibility of Pk).The peak set Pk is shuffle-compatible, and the linear map on A Pk defined by π Pk → K |π|,Pk π is a Q-algebra isomorphism from A Pk to Π.
The analogue of Stembridge's quasisymmetric peak functions in the cyclic setting are the cyclic peak quasisymmetric functions K cyc n,S recently introduced by Liang [Lia22].Here, we shall define the cyclic peak functions K cyc n,S in terms of the K n,S .For brevity, let us say that S is a cyclic peak set of [n] if S is the cyclic peak set of some permutation of length n.Then, if S is a cyclic peak set of [n], let where π is any permutation in P n with cyclic peak set S. We can also write since the K cyc n,S are invariant under cyclic shift.Liang showed that the K cyc n,[S] are linearly independent, and they span a subalgebra Λ of cQSym called the algebra of cyclic peaks. 4he following theorem-which is equivalent to Equation (5.10) of [Lia22]-gives a multiplication rule for the K cyc n,[S] .This multiplication rule also implies that cPk is cyclic shufflecompatible, which was first proven by Domagalski et al. [DLM + 21] using bijective means.
Theorem 4.5.Let m and n be non-negative integers, let A be a cyclic peak set of [m], and let B be a cyclic peak set of where Proof.First, we take φ Pk : QSym → Π to be the composition of the map F L → π Pk with the map π Pk → K |π|,Pk π from Theorem 4.4 where π is any permutation with Pk π = Pk L; then φ Pk satisfies the conditions in Theorem 3.4.
Let S be a non-Escher subset of [n], and let [P ] be the cyclic peak set of any cyclic permutation [π] of length n with cyclic descent set [S].Note that the sets (S + i) ∩ [n − 1] where i ranges from 1 to n are precisely the descent sets of the n linear permutations in [π].Hence, we have Clearly, φ cPk (F cyc n,S ) depends only on the cPk-equivalence class of the cyclic composition cComp[S], and we know that the K cyc n,[P ] are linearly independent.Applying Theorem 3.7, we conclude that cPk is cyclic shuffle-compatible and that , from which the multiplication rule (2) follows.
Corollary 4.6 (Cyclic shuffle-compatibility of cPk).The cyclic peak set cPk is cyclic shufflecompatible, and the linear map on A cyc cPk defined by 4.3.The cyclic shuffle algebra of (cpk, cdes).We will now use Theorem 3.7 to construct the cyclic shuffle algebra A cyc (cpk,cdes) from the linear shuffle algebra A (pk,des) .We begin by recalling the following result about A (pk,des) , which is Theorem 5.9 of Gessel and Zhuang [GZ18].Below, we will use the notation Q[[t * ]] to denote the Q-algebra of formal power series in t where the multiplication is given by the Hadamard product * , defined by Theorem 4.7 (Shuffle-compatibility of (pk, des)).
Then the linear map on A (pk,des) defined by We note that, in the definition of u , all products should be interpreted as ordinary multiplication; the Hadamard product in t is only used when multiplying elements in the span of the u (pk,des) n,j,k .The same is true in Theorems 4.8, 4.9, and 4.10 presented later in this section.
Then the linear map on A cyc (cpk,cdes) defined by (cpk,cdes) has dimension ⌊n 2 /4⌋.Proof.We shall apply Theorem 3.7 using st = (pk, des).In doing so, we take φ (pk,des) to be the composition of the map F L → π (pk,des) with the map from Theorem 4.7 (b), where π is any permutation with pk π = pk L and des π = des L.
Let π be a permutation of length n ≥ 2 with cyclic descent set S, and let j = cpk[π] and k = cdes[π] (which only depend on S and not the specific choice of π).Let us consider the n linear permutations in [π], whose descent sets are given by (S + i) ∩ [n − 1] where i ranges from 1 to n.Among these n permutations, the following hold: • Exactly j of these permutations have cpk .
For n = 0 and n = 1, we have Clearly, φ (cpk,cdes) (F cyc n,S ) depends only on the (cpk, cdes)-equivalence class of cComp[S].To prove linear independence, let us order monomials in the variables t and y lexicographically by the exponent of t followed by the exponent of y, that is, t a y b > t c y d if and only if either a > c, or if a = c and b > d.Since Corollary 4.3 implies j ≥ 1, it is readily verified that the least monomial in (1 x n v (cpk,cdes) n,j,k 1≤j≤⌊n/2⌋ j≤k≤n−j is linearly independent for each n ≥ 2, and this in turn implies that 1≤j≤⌊n/2⌋ j≤k≤n−j is linearly independent.Corollary 4.3 ensures that we have the correct limits on j and k, so we can use Theorem 3.7 to conclude that parts (a) and (b) hold.From Corollary 4.3, we know that for n ≥ 2, the number of (cpk, cdes)-equivalence classes of cyclic permutations of length n is and it is straightforward to show that this is equal to ⌊n 2 /4⌋.Thus, part (c) follows.
4.4.The cyclic shuffle algebras of cpk and cdes.Next, we use our characterization of the cyclic shuffle algebra A cyc (cpk,cdes) along with Theorem 2.1 to characterize A cyc cpk and A cyc cdes , which also provides an alternative proof for the cyclic shuffle-compatibility of cpk and cdes.
Let N be the set of non-negative integers.In the theorems below, we use the notation Q[x] N to denote the algebra of functions N → Q[x] in the non-negative integer variable p.For example, the map p → p 2 x + p 3 -which we write simply as p 2 x + p 3 for brevity-is an element of Q[x] N .Moreover, in Theorem 4.9 below, n k is the number of k-element multisubsets of [n].
(a) The cyclic peak number cpk is cyclic shuffle-compatible.
(b) The linear map on A cyc cpk defined by Then the linear map on A cyc cpk defined by be the composition of the map from Theorem 4.8 (b) and the y = 1 evaluation map.Since x n for all n ≥ 1, we see that φ is precisely the map in part (b) of this theorem.Note that v | y=1 depends only on n and j, so the v | y=1 correspond to cpk-equivalence classes.Furthermore, it is straightforward to verify that the v | y=1 are linearly independent, so we may apply Theorem 2.1 to complete the proof for parts (a), (b) and (d).Part (c) follows from part (b) and the identity where the first equality follows from [Lia22, Proposition 5.13 and Corollary 5.18].
(a) The cyclic descent number cdes is cyclic shuffle-compatible.
(b) The linear map on A cyc cdes defined by The linear map on A cyc cdes defined by The proofs for parts (a), (b), and (d) follow in the same way as in for Theorem 4.9, except that we evaluate at y = 0 as opposed to y = 1.Part (c) follows from part (b) and the identity which was established in [AGRR21, Lemma 5.8].

Cyclic permutation statistics induced by linear permutation statistics
Recall that the cyclic permutation statistics cDes and cPk are defined by In other words, cDes[π] is simply the distribution of the linear permutation statistic cDes over all linear permutations in [π], and similarly with cPk[π].In fact, any linear permutation statistic st induces a multiset-valued cyclic permutation statistic (which we also denote st by a slight abuse of notation) if we let In this section, we study these multiset-valued cyclic statistics induced from various linear permutation statistics.
5.1.The cyclic statistics Des, des, Pk, and pk.To begin, we note that the cyclic statistics induced from the linear statistics Des, des, Pk, and pk are equivalent to cDes, cdes, cPk, and cpk, respectively.Conversely, suppose that we are given Pk[π] and wish to recover cPk[π].Let i ∈ [n] be arbitrary.Notice that, among all n representatives of [π], the index of π i spans the entire range {1, 2, . . ., n}.If i is a cyclic peak of π in particular, this means that the index of π i will be a peak of all n representatives of [π] except for the linear permutation beginning with π i and the one ending with π i ; hence, if one adds up pk π over all π ∈ [π], then each of these π i will contribute n − 2 to the summation.It follows that the sum of the sizes of all peak sets in Pk Since cDes, cdes, cPk, and cpk are cyclic shuffle-compatible, it follows from these equivalences and Theorem 2.2 that the cyclic statistics Des, des, Pk, and pk are as well.
Theorem 5.5 (Cyclic shuffle-compatibility of Des, des, Pk, and pk).The cyclic statistics Des, des, Pk, and pk are cyclic shuffle-compatible, and we have the Q-algebra isomorphisms , and A cyc pk ∼ = A cyc cpk .5.2.Symmetries revisited.Let f be a length-preserving involution on permutations that is both shuffle-compatibility-preserving and rotation-preserving.In Section 2.3, we proved that if the cyclic permutation statistics cst 1 and cst 2 are f -equivalent and if cst 1 is cyclic shuffle-compatible, then cst 2 is also cyclic shuffle-compatible with cyclic shuffle algebra isomorphic to that of cst 1 .We now show that f -equivalence of two linear permutation statistics induces f -equivalence of their induced cyclic statistics.
Lemma 5.6.Let f be rotation-preserving.If st 1 and st 2 are f -equivalent linear permutation statistics, then their induced cyclic permutation statistics st 1 and st 2 are f -equivalent.
Theorem 5.7.Let f be shuffle-compatibility-preserving and rotation-preserving, and let st 1 and st 2 be f -equivalent linear permutation statistics.If the induced cyclic statistic st 1 is cyclic shuffle-compatible, then the induced cyclic statistic st 2 is also cyclic shuffle-compatible and A cyc st 2 is isomorphic to A cyc st 1 .Proof.This is an immediate consequence of Theorem 2.6 and Lemma 5.6.
Corollary 5.8.Suppose that the linear permutation statistics st 1 and st 2 are r-equivalent, c-equivalent, or rc-equivalent.If the induced cyclic statistic st 1 is cyclic shuffle-compatible, then the induced cyclic statistic st 2 is also cyclic shuffle-compatible and its cyclic shuffle algebra A cyc st 2 is isomorphic to A cyc st 1 .Given π ∈ P n , recall that the valley set Val statistic is defined by and let us also define the cyclic valley set cVal by As a sample application of Corollary 5.8, observe that Val is c-equivalent to Pk (as linear permutation statistics) and similarly with cVal and cPk.Combining this with Lemma 5.3, we immediately obtain the following.Theorem 5.9 (Cyclic shuffle-compatibility of Val and cVal).The cyclic statistics Val and cVal are cyclic shuffle-compatible, and we have the Q-algebra isomorphisms

Cyclic major index.
A natural question to ask is whether there is a nice cyclic analogue of the major index.This question was raised in [AGRR21]  From Stanley's theory of P -partitions [Sta72], one gets the q-analogue (3) τ ∈π¡σ q maj τ = q maj π+maj σ m + n m where m+n m is a q-binomial coefficient.Note that (3) implies that maj is shuffle-compatible.
It can be shown that , so one could ask that the cyclic major index give a q-analogue of this identity, similar to (3), or at least for the cyclic major index to be cyclic shuffle-compatible.
Stanley also refined Equation (3) as follows.Let If des π = i and des σ = j, then (4) in particular, this implies that (des, maj) is shuffle-compatible, and so we would like a cmaj statistic for which cmaj and (cdes, cmaj) are both cyclic shuffle-compatible.
In [AGRR21], Adin et al. computed the cardinality of which inspired Ji and Zhang [JZ22] to define a cmaj statistic which gives a q-analogue of this count.They proved a generating function formula analogous to (4), but unfortunately, the formula does not simplify into single product, and one could hope for a different cyclic major index whose generating function would do so.Furthermore, their formula does not actually show that their (cdes, cmaj) is cyclic shuffle-compatible; in fact, neither of their cmaj and (cdes, cmaj) are cyclic shuffle-compatible.Each of the cyclic statistics cDes, cdes, cPk, and cpk is (or is equivalent to) a multisetvalued cyclic statistic induced by a corresponding linear permutation statistic, so a natural alternative definition for a cyclic major index would be to define cmaj first on linear permutations and then consider the multiset-valued statistic induced by the linear cmaj.To that end, given a linear permutation π, let cmaj π := k∈cDes π k.
Proof.Let us assume throughout this proof that n ≥ 2, as the cases n = 0 and n = 1 are trivial.To see that cDes is a refinement of ocmaj, suppose cDes[π] = cDes[σ] where π and σ have the same length n.So, we can write [σ] = {σ (1) , σ (2) , . . ., σ (n) } where cDes π For the converse, it is sufficient to show that the cyclic descent composition cComp[π] can be reconstructed from ocmaj [π].First, recall Equation (5): Since n ≥ 2, we have 1 ≤ cdes[π] ≤ n − 1, and together with the above equation, we have that n ∈ cDes π (i) if and only if cmaj π (i) > cmaj π (i+1) .A similar argument shows that we can never have cmaj π (i) = cmaj π (i+1) .Now, suppose we are given ocmaj[π] = [m 1 , m 2 , . . ., m n ] where m i = cmaj π (i) .Let s and t be two consecutive cyclic descents of ocmaj[π], i.e., where subscripts are considered modulo n as usual.From the previous paragraph, it follows that n is in both cDes π (s) and cDes π (t) , and that the penultimate descent in cDes π (s) becomes the descent n ∈ cDes π (t) with n never being a descent for any of the intermediate cyclic descent sets.So t − s (modulo n) is a part of the cyclic composition cComp [π].Therefore, all the parts of cComp[π] can be determined, and their order will be the same as that induced by the consecutive cyclic descents in ocmaj[π].Thus we have reconstructed cComp[π] from ocmaj[π], completing the proof.
Corollary 5.13 (Cyclic shuffle-compatibility of ocmaj).The ordered cyclic major index ocmaj is cyclic shuffle-compatible, and its cyclic shuffle algebra A cyc ocmaj is isomorphic to A cyc cDes .Of course, one could wonder if the unordered multiset of cmaj values is also equivalent to cDes for cyclic permutations, but this is not the case.Indeed, if the cyclic permutation statistics cmaj and cDes were equivalent, then the cyclic shuffle-compatibility of cDes would imply that cmaj is cyclic shuffle-compatible as well, which we know to be false.5.4.Other descent statistics.To conclude this section, let us consider the cyclic permutation statistics induced by the following linear descent statistics: • The valley number val, which we defined earlier to be the number of valleys of a permutation.• The double descent set Ddes and the double descent number ddes.We call i ∈ {2, 3, . . ., n − 1} a double descent of π ∈ P n if π i−1 > π i > π i+1 .Then Ddes π is the set of double descents of π, and ddes π the number of double descents of π. • The left peak set Lpk and the left peak number lpk.We call i ∈ [n − 1] a left peak of π ∈ P n if i is a peak of π, or if i = 1 and π 1 > π 2 .Then Lpk π is the set of left peaks of π, and lpk π the number of left peaks of π. • The right peak set Rpk and the right peak number rpk.We call i ∈ {2, 3, . . ., n} a right peak of π ∈ P n if i is a peak of π, or if i = n and π n−1 < π n .Then Rpk π is the set of right peaks of π, and rpk π the number of right peaks of π. • The exterior peak set Epk and the exterior peak number epk.We call i ∈ [n] an exterior peak of π ∈ P n if i is a left peak or right peak of π.Then Epk π is the set of exterior peaks of π, and epk π the number of exterior peaks of π. • The number of biruns br and the number of up-down runs udr.A birun of π is a maximal consecutive monotone subsequence of π; an up-down run of π is a birun of π, or the first letter π 1 of π if π 1 > π 2 .Then br π and udr π are the number of biruns and the number of up-down runs, respectively, of π.For example, take π = 713942658.Then we have val π = 3, Ddes π = {5}, ddes π = 1, Lpk π = {1, 4, 7}, lpk π = 3, Rpk π = {4, 7, 9}, rpk π = 3, Epk π = {1, 4, 7, 9}, epk π = 4, br π = 6, and udr π = 7.
Aside from Ddes, ddes, and br, all of the above statistics (as linear permutation statistics) are shuffle-compatible.Also, because these are all descent statistics, each of the induced cyclic statistics are cyclic descent statistics.Indeed, if we are given cDes[π] and the length of π, then we can determine Des Because des and cdes are equivalent as cyclic permutation statistics and similarly with pk and cpk, one might expect the cyclic statistics br and cbr to be equivalent as well, but this is not the case because br is not actually cyclic shuffle-compatible.For instance, consider π = 25673489, σ = 24567389, and ρ = 1.Then br[π] = br[σ], but the multiset {{5 4 , 6 4 , 7}} appears four times in br([π] ¡[ρ]) but only twice in br([σ] ¡[ρ]).One can also use the same permutations π, σ, and ρ to show that (br, des) is not cyclic shuffle-compatible.
Even though the linear statistics Lpk and Epk are shuffle-compatible, their induced cyclic statistics are not cyclic shuffle-compatible.As a counterexample, take Observe that Rpk is r-equivalent to Lpk and rpk is r-equivalent to lpk.Hence, by Corollary 5.8, neither Rpk nor rpk are cyclic shuffle-compatible.One can also define "left", "right", and "exterior" versions of the valley set and valley number; by similar symmetry arguments, none of these are cyclic shuffle-compatible either.
In contrast, the exterior peak number epk and the pair (epk, des) are cyclic shufflecompatible because they are equivalent to cpk and (cpk, cdes), respectively.To prove these equivalences, we will also need to consider the cyclic valley number statistic cval: we say that i ∈ [n] is a cyclic valley of π ∈ S n if π i−1 > π i < π i+1 with the indices considered modulo n, and cval[π] is defined to be the number of cyclic valleys of any permutation in [π].Equivalently, cval[π] is the cardinality of the cyclic valley set cVal[π] defined in Section 5.2.
Lemma 5.15.The cyclic permutation statistics val and cval are equivalent.
It would be interesting to find new cyclic shuffle-compatibility results stemming from these statistics-i.e., if one of the ost is cyclic shuffle-compatible and is not equivalent to another statistic already known to be shuffle-compatible.On the other hand, it would also be interesting to find nontrivial equivalences between these statistics and others, regardless of whether they are cyclic shuffle-compatible.
Next, we pose a question related to the lifting lemma of Domagalski et al. [DLM + 21, Lemma 2.3], which provides an avenue for proving cyclic shuffle-compatibility of a cyclic descent statistic using the shuffle-compatibility of a related linear descent statistic.The lifting lemma involves two maps S i and M, defined as follows.Given π ∈ S n and i ∈ We would like to understand how the lifting lemma fits into our algebraic framework.In particular, we have tried to prove the lifting lemma from Theorem 2.8, but our attempts have been unsuccessful because it is unclear to us how the conditions in the lifting lemma relate to the linear independence condition of that theorem.Question 6.3.Can the lifting lemma be proven from Theorem 2.8?Finally, every statistic which is known to be cyclic shuffle-compatible is a cyclic descent statistic, so it is natural to ask whether any cyclic shuffle-compatible statistics are not cyclic descent statistics.In the linear setting, Gessel and Zhuang [GZ18] had conjectured that every shuffle-compatible statistic is a descent statistic, but a counterexample was found by Kantarcı Oguz [KO22].So, we will pose this as a question rather than as a conjecture.Question 6.4.Is every cyclic shuffle-compatible statistic a cyclic descent statistic?
We note that the cyclic statistic induced by Kantarcı Oguz's counterexample is not cyclic shuffle-compatible.
− denote the span of {F cyc n,[S] } over all n ≥ 0 and all equivalence classes [S] of non-Escher subsets S ⊆ [n].The following theorem, proven by Adin et al. [AGRR21, Theorem 3.22], gives a multiplication rule for the fundamental cyclic quasisymmetric functions in cQSym − , which also implies that the cyclic descent set cDes is cyclic shuffle-compatible and has cyclic shuffle algebra isomorphic to cQSym − .Theorem 3.2.Let m and n be non-negative integers, and let A ⊆ [m] and B ⊆ [n] be non-Escher subsets.Then (1) [π] is any cyclic permutation of length m with cyclic descent set [A] and [σ] is any cyclic permutation (with σ disjoint from π) of length n with cyclic descent set [B]. Adin et al. proved Theorem 3.2 using toric [ D]-partitions; we now supply an alternative proof using Theorem 2.8.
[π] is any cyclic permutation of length m with cyclic peak set [A] and [σ] is any cyclic permutation (with σ disjoint from π) of length n with cyclic peak set [B].While Liang's proof of Theorem 4.5 uses enriched toric [ D]-partitions, we shall now use Theorem 3.7 to supply an alternative proof.
and again in [DLM + 21].One first needs to explain what one means by "nice."If π ∈ P m and σ ∈ P n , then |π ¡ σ| = m + n m .