$P$-partitions and $p$-positivity

Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.


Introduction
Whenever a new family of symmetric functions is discovered, one of the most logical first steps to take is to expand them in one of the many interesting bases of the space of symmetric functions.This paradigm can be traced from Newton's identities to modern textbooks such as [Mac79].Of special interest are expansions in which all coefficients are nonnegative integers.Such coefficients frequently encode highly nontrivial combinatorial or algebraic information.
One of the most well-studied bases is formed by the power sum symmetric functions.Symmetric functions that expand into power sum symmetric functions with nonnegative coefficients are called p-positive.Recent works in which p-positivity is discussed include [SW10, SSW11, Ath15, SW16, Ell16, AP17].The expansion of a symmetric function into power sum symmetric functions can be useful, for instance, when one is working with plethystic substitution [LR10], or evaluating certain polynomials at roots of unities [Dé83,SSW11].Suppose X is a symmetric function for which we would like to know the expaonsion into power sum symmetric functions.In some of the papers mentioned above the following pattern recurs.First, expand X into fundamental quasisymmetric functions using R. Stanley's theory of P -partitions.Secondly, conduct some analysis specific to the function X at hand to obtain the p-expansion of X. Ideally, one would ask for a more uniform approach.Question 1.Is there a uniform method for deriving the expansion of a given symmetric function into power sum symmetric functions whenever the theory of P -partitions is applicable?
In practice the functions of interest often belong to the larger space of quasisymmetric functions.Clearly if a quasisymmetric function expands into power sum symmetric functions, positively or not, then it has to be symmetric.This leads to results of the following type: "Suppose X belongs to some special family of quasisymmetric functions F. Then X is p-positive if and only if X is symmetric."In this case it is very natural to ask the following.Question 2. Is there a more general positivity phenomenon hiding in the background, which encompasses all quasisymmetric functions that belong to F ?
In this paper we answer both Question 1 and Question 2 in the affirmative.Key to these answers are the quasisymmetric power sums Ψ α .Quasisymmetric power sums originate in the work of I. Gelfand et al. [GKL + 95] on noncommutative symmetric functions, and were recently investigated by C. Ballantine et al. [BDH + 17]. 1 The family Ψ α , where α ranges over all compositions of n, forms a basis of the space of homogeneous quasisymmetric functions of degree n, and refines the power sum symmetric functions.This paper has two main results that easily fit into the existing theory of quasisymmetric functions.The first result is a formula for the expansion of the fundamental quasisymmetric functions into quasisymmetric power sums.
Theorem 1 (Theorem 3.1).Let n ∈ N and S ⊆ [n − 1].Then the fundamental quasisymmetric function F n,S expands into quasisymmetric power sums as where the sum ranges over all compositions α of n such that S is α-unimodal.
Here we use the standard notation [n] := {1, 2, . . ., n}.The definitions of αunimodal sets and the set S α are found in Sections 2. The quasisymmetric functions F n,S and Ψ α and the factor z α are defined in Section 3. The proof of Theorem 1 relies on a new result on α-unimodal sets, and on the hook-length formula for forests.
The second main result concerns reverse P -partitions, which were introduced by R. Stanley [Sta72].In the simplest case reverse P -partitions are order-preserving maps from a finite poset P to the positive integers.The generating function of reverse P -partitions is defined as 1 There are several different quasisymmetric power sum bases.We use one denoted by Ψα in The function K P is a homogeneous quasisymmetric function of degree n = |P |.We prove that K P expands positively into quasisymmetric power sums Ψ α and provide two combinatorial interpretations for the involved coefficients.
Theorem 2 (Theorems 4.2 and 5.4).Let (P, w) be a naturally labeled poset with n elements.Then were both sums range over all compositions α of n.In particular, the quasisymmetric function K P is Ψ-positive.
The set L * α (P, w) consists of certain α-unimodal linear extensions 2 of P .The definition is given in Section 2. The set O * α (P ) consists of certain order-preserving surjections from P onto a chain.The definition is given in Section 5.The proof of Theorem 2 uses the well-known expansion of K P into the fundamental basis, Theorem 1, and a sign-reversing involution closely related to an involution constructed by B. Ellzey in [Ell16,Thm. 4.1].
It follows from Theorem 2 that any symmetric function which is a positive linear combination of functions K P for posets P is p-positive.This affirms Question 1.
It is a manifestation of the ubiquity of reverse P -partitions in algebraic combinatorics that many interesting families of symmetric and quasisymmetric functions can be expressed as nonnegative linear combinations of functions K P .By Theorem 2 each function X that belongs to such a family is Ψ-positive.This answers Question 2 for a large class of families F.
As an application we give positivity results and combinatorial interpretations for the coefficients in the expansion into (quasi)symmetric power sums for the following families of quasisymmetric functions: • The chromatic quasisymmetric functions of J. Shareshian and M. Wachs, [SW16].We prove a generalization of a recent result by B. Ellzey [Ell16], that applies to all directed graphs, and not only those with a symmetric chromatic quasisymmetric functions.Our result also extends to a q-generalization of so called k-balanced chromatic quasisymmetric functions that were introduced by B. Humpert in [Hum11].• Unicellular and vertical-strip LLT polynomials, which are of special interest in [CM17] and in the study of diagonal harmonics.This generalizes an observation in [AP17,HW17] and answers an open problem in [AP17].Furthermore, this result provides more supporting evidence regarding a related e-positivity conjecture.• The multivariate Tutte polynomials introduced by R. Stanley [Sta98], and the more general B-polynomals on directed graphs due to J. Awan and O. Bernardi, [AB16].• The quasisymmetric functions associated to matroids due to L. Billera, N. Jia and V. Reiner, [BJR09].
Sym. monomial m λ (x) Powersum p λ (x) Schur s λ (x) Complete h λ (x) QSym.monomial M α (x) QSym.powersum Ψ α (x) Fundamental F α (x) Figure 1.An overview of the families of functions we discuss.The shaded families are symmetric functions (and bases for the corresponding space).The remaining families are bases for the space of quasisymmetric functions, except for K P which is too large to be a basis.The arrows represent the relation "expands positively in" (which of course is a transitive relation).The dashed line is the result in Theorem 2.
Figure 1 gives an overview of some of the bases of symmetric and quasisymmetric functions that are mentioned in this paper.
1.1.Outline.In Section 2 we engage in the combinatorics of α-unimodal permutations and sets.We prove two new results which are instrumental in the proofs of our main theorems.In particular we define the set L * α (P, w) attached to a labeled poset as the set of certain α-unimodal linear extensions of P .In Section 3 we give a short introduction to quasisymmetric functions and define quasisymmetric power sums.We proceed to prove Theorem 1 and conclude [Ath15, Prop.3.2] as a corollary.In Section 4 we define reverse P -partitions and prove the first half of Theorem 2. Section 5 is dedicated to order-preserving surjections onto chains.It contains the definition of the set O * α (P ) and the proof of the second half of Theorem 2. In Section 6 we generalize Theorem 2 to include weighted posets or, equivalently, reverse P -partitions with forced equalities.This is perhaps the most technical section and it is not required to understand the rest of the paper.In Section 7 we use the developed tools to derive Ψ-expansions of some of the most commonly used bases of the space of symmetric functions (h λ , p λ and s λ ), including Roichman's formula [Roi97,Thm. 4].Moreover we obtain the positivity results mentioned above.Finally, in Section 8 we mention several interesting direction that could be pursued in the future, as well as some ideas that, sadly, do not work.1.2.Acknowledgements.The authors would like to thank Svante Linusson.The first author is funded by the Knut and Alice Wallenberg Foundation (2013.03.07).

α-unimodal combinatorics
In this section we investigate α-unimodal permutations, sets and compositions.Our main objective is to prove two bijective results, namely Theorem 2.5 and Theorem 2.9, which we apply to the theory of quasisymmetric functions in the subsequent sections.However, we contend that α-unimodal combinatorics is an interesting topic in its own right.
The so called α-unimodal sets, where α is a composition, first appear in a recursive formula for Kazhdan-Lusztig characters of the Hecke algebra of type A n−1 due to Y. Roichman [Roi97,Thm. 4].A bijective treatment of this formula was later given by A. Ram [Ram98].The term α-unimodal was coined in [AR15].We refer to [ER13,Thi01] for more results on unimodal permutations.More recently α-unimodal sets were used in the works of C. Athanasiadis [Ath15] and B. Ellzey [Ell16] to derive the power sum expansions of certain families of symmetric functions. (3) For instance, note that a permutation σ ∈ S n is unimodal if and only if the set . We remark that this definition, which is borrowed from [AR15, Ath15, Ell16], is not standard in the study of unimodal sequences, where one would usually assume is more natural if one wants to use descents rather than ascents, which is common practice when working with tableaux and quasisymmetric functions.In Sections 7 and 8 we discuss unimodality of polynomials.In that case the standard definition is used, that is, a polynomial a 0 + a 1 q Given a composition α of n with parts define The correspondence α → S α is a well-known bijection between compositions of n and subsets of [n − 1].Sometimes it is more convenient to work with the set S 0 α := S α ∪ {0} instead.Let α, β be compositions of n.We say α refines β if S β ⊆ S α .Refinement on compositions is denoted by α ≤ β.Equivalently, α ≤ β if β can be obtained by adding contiguous parts of α.We also say β is coarser than α.
Define the blocks of α as the sets Let α be a composition of n.A permutation σ ∈ S n is α-unimodal if the word obtained by restricting σ to the block The descent set of a permutation σ ∈ S n is defined as There is yet another equivalent description of α-unimodal sets.Define the binary sequence a 1 , . . ., a n−1 by letting a i = 1 if i ∈ S, and a i = 0 otherwise.Similarly define b 1 , . . ., b n−1 by b i = 1 if i ∈ S α , and b i = 0 otherwise.Then S is α-unimodal if and only if the two-line arrangement In short, if 2 appears, then 1 must also appear and, similarly, if 5 appears then 4 must also be there.Equivalently, if the two-line arrangement a 1 a 2 a 3 a 4 a 5 0 0 1 0 0 avoids the forbidden pattern, then a 2 = 1 implies a 1 = 1, and a 5 = 1 implies a 4 = 1.

2
We start out by collecting some basic combinatorial facts on α-unimodality.All of them are straightforward to prove and should be viewed as a warm up to get acquainted with α-unimodal sets.
Proposition 2.3.Let n ∈ N and α, β be compositions of n with parts.
The set U α is closed under unions and intersections and therefore forms a sublattice of the Boolean lattice.
(vii) The lattice U α is the direct product of chains.Its Möbius function is given by where m denotes the number of indices i The set V α defined in (viii) is an order ideal and thus a meet-semilattice in the lattice of compositions ordered by refinement.However, V α is generally not a sublattice.(x) The q, t-generating function of α-unimodal sets is given by Proof.Claims (i), (ii), (iii), (iv) and (vi) follow directly from the definition of α-unimodal sets.Claim (ix) is an immediate consequence of Claim (ii).
which yields (v).In fact, the map ϕ is an isomorphism of partially ordered sets, where by definition (T, r 1 , . . ., r ) ≤ (T , r 1 , . . ., r ) if and only if T ⊆ T and r i ≤ r i for all i ∈ [ ]. Thus Claim (vii) follows from standard techniques for computing the Möbius function of finite posets, see [Sta11,Sec. 3.8].
The remaining two claims are best understood using the definition of α-unimodality via two-line arrangements of zeroes and ones.To see Claim (viii) let a 1 , . . ., a n−1 be a fixed binary sequence encoding the set S α .We are looking to determine the number of compatible sequences b 1 , . . ., b n−1 .Each part of α except the last part corresponds to a pattern a i a i+1 = 01 if it is greater than 1.There are therefore three choices for b i b i+1 , namely 01, 10 or 11.In total this contributes the factor of 3 m in (5).All other entries of b can be chosen arbitrarily, contributing a factor of 2 n−2m−1 .
In order to see Claim (x) note that a pair of compatible sequences a, b ∈ {0, 1} n is either empty, or it can be obtained from a pair of shorter compatible sequences a , b ∈ {0, 1} n−1 by appending one of the four patterns 0 0 Sequences a, b that are obtained in this way and do contain the forbidden pattern are of the form a 0 1 b 0 0 where a , b ∈ {0, 1} n−2 avoid the forbidden pattern.Thus we conclude the recursion F (q, t, z) = zt + (1 + q + t + qt)zF (q, t, z) − qz 2 F (q, t, z) and Claim (x) follows.
The relation defined on the subsets of [n − 1] by letting S ≺ S α if and only if S is α-unimodal is neither symmetric, antisymmetric nor transitive.It follows from Proposition 2.3 (x) that the total number f (n) of such relations satisfies the recursion f (n) = 4f (n − 1) − f (n − 2) with f (0) = 0 and f (1) = 1, and is therefore equal to the sequence A001353 in [Slo16].This relates α-unimodal sets to, for example, spanning trees in a 2 × n grid.
It is also easy to obtain q-analogues of Proposition 2.3 (v) and (viii) -this is left as an exercise.
In order to state the first main result of this section we need one more definition, which is due to C. Ballantine et al. and appears in the study of quasisymmetric analogues of the power sum symmetric functions [BDH + 17].
Let α ≤ β be compositions of n.Given a permutation σ ∈ S n and i ∈ [ (α)] define the subword the subwords σ (i) , . . ., σ (j) are sorted increasingly with respect to their maximal elements, where i, j ∈ [ (α)] are determined by Let because the maxima in the third β-block are not in increasing order. 2 The definitions of both α-unimodal permutations and consistent permutations are somewhat out of the blue at first glance, however, these two concepts interplay in an interesting fashion.The following theorem will allow us to expand Gessel's fundamental basis into quasisymmetric power sums in Section 3. Theorem 2.5.Let n ∈ N and β, γ be compositions of n.Then where the sum ranges over all compositions α of n such that α ≤ β and S γ is α-unimodal.
Proof.Let R(β, γ) denote the set of all compositions α of n such that α ≤ β and S γ is α-unimodal.
Case 1.First assume β ≤ γ.Then S γ is α-unimodal for all α ≤ β by Proposition 2.3 (iii).Moreover S γ \ S α = ∅ for all α ∈ R(β, γ).Thus it suffices to give a bijection ϕ : This is accomplished simply by reversing each subword σ (i) .The same idea appears in the well-known method for switching between cycle notation and one line notation by forgetting the parentheses, see [Sta11,Sec. 1.3].Note that ϕ −1 depends on β.
by minimality of i.Thus for S γ to be α-unimodal we must have i − 1 ∈ S α or i ∈ S α by Proposition 2.3 (i).In fact, the set R(β, γ) is partitioned into sets of three, {α, α , α }, such that We claim that there exists a bijection Define the map ϕ as follows.For σ ∈ Cons(α, β) set With the notation from Example 2.4 this amounts to the following: The claim follows from We now turn to labeled posets, for which we adopt the same conventions as in [Sta11].A labeled poset (P, w) is a finite poset P equipped with a bijection w : P → [n].We call (P, w) a naturally labeled poset if w is order-preserving, that is, w(x) < w(y) for all x, y ∈ P with x < P y.
The Jordan-Hölder set of a labeled poset (P, w) with n elements is defined as That is, σ ∈ L(P, w) if and only if σ −1 • w is a linear extension of P .
To avoid ambiguity we refer to the values w(x), where x ∈ P , as labels.Other functions f : P → N (such as σ −1 •w, where σ ∈ L(P, w)) we sometimes call colorings and their values f (x) colors.With this convention the elements σ ∈ L(P, w) map colors to labels.
Example 2.6.For example, let (P, w) be the labeled poset below: Given a labeled poset (P, w) with n elements and a composition α of n with parts, let L α (P, w) := {σ ∈ L(P, w) : σ is α-unimodal}.Furthermore, given σ ∈ S n , define the subposets Finally denote by L * α (P, w) the subset of L α (P, w) that consists of the elements σ ∈ L α (P, w) such that P α i (σ) contains a unique minimal element for all i ∈ [ ]. Example 2.7.Let (P, w) be as in Example 2.6.If α = (2, 3), then The set L * α (P, w) associated to a naturally labeled poset, turns out to be a highly useful concept in the study of power sum symmetric functions.The following lemma gives a necessary condition for a permutation σ to lie in L * α (P, w).Lemma 2.8.Let (P, w) be a naturally labeled poset with n elements, let α be a composition of n, and let σ ∈ L * α (P, w).Then DES(σ) ⊆ S α .
We now come to the second main result of this section.The following theorem will be used to prove that several families of quasisymmetric or symmetric functions expand positively into power sums.
Theorem 2.9.Let (P, w) be a naturally labeled poset with n elements, and let α be a composition of n.Then In particular, the left-hand side of (8) is nonnegative.
Proof.We prove the theorem by the use of a sign-reversing involution Let be the number of parts of α, and let σ ∈ L α (P, w) \ L * α (P, w).Then there exists a minimal index i ∈ [ ] such that P α i (σ) has at least two minimal elements.Suppose In particular, M = σ j or M = σ m .We distinguish between these two cases.
It is straightforward to verify that ϕ(σ) is α-unimodal in both cases.Moreover, P α r (σ) = P α r (ϕ(σ)) for all r ∈ [ ], so by the definition of x ∈ P α r (σ) and y ∈ P α s (σ) with r < s.To show that ϕ is well-defined, it therefore suffices to verify that the restriction of ϕ(σ) −1 • w to P α i (σ) is a linear extension.Suppose we are in Case 1 and set x = w −1 (M ).In order to prove ϕ(σ) ∈ L(P, w) we need to verify ϕ(σ) −1 • w(x) < ϕ(σ) −1 • w(y) for all y ∈ P α i (σ) with x < P y.Thus assume that y ∈ P α i (σ) satisfies x < P y.Then M = w(x) < w(y) implies σ −1 • w(y) > j.By the defining property of j and m we also have σ −1 • w(y) > m, and therefore Next assume we are in Case 2 and set y = w −1 (M ).To show that ϕ(σ) ∈ L(P, w) we need to verify But this is trivially true because y is a minimal element of P α i (σ).To see that ϕ is an involution note that M > σ k since P α i (σ) has at least two minimal elements by assumption.Thus σ belongs to Case 1 if and only if ϕ(σ) belongs to Case 2.
It is also clear that ϕ is sign-reversing.Indeed whenever σ belongs to Case 1, and equivalently, The claim in (17) now follows from Lemma 2.8, which guarantees that DES(σ) ⊆ S α for all σ ∈ L * α (P, w).

Quasisymmetric functions
The main result of this section, Theorem 3.1, is the expansion of Gessel's fundamental basis into quasisymmetric power sums.As an immediate consequence we obtain a new proof of a recent result of C. Athanasiadis [Ath15, Prop.3.2] (Corollary 3.4 below).We start out with a brief introduction to quasisymmetric functions.For more background the reader is referred to [Sta01,LMvW13].
such that the degree of f is finite, and for every composition (α 1 , . . ., α ) the coefficient of The functions M α , where α ranges over all compositions of n, constitute a basis for the space of homogeneous quasisymmetric functions of degree n.
Another basis for the space of quasisymmetric functions are the fundamental quasisymmetric functions.The fundamental quasisymmetric functions of degree n are indexed by subsets S ⊆ [n − 1] and defined as Alternatively, given a composition α of n, we sometimes write F α := F n,Sα .The expansion of fundamental quasisymmetric functions into monomial quasisymmetric functions is given by Given a composition α set z α := i≥1 i mi m i !, where m i denotes the number of parts of α that are equal to i.The following definitions appear in where α (i) is the composition of β i that consists of the parts α j with B α j ⊆ B β i .The quasisymmetric power sum Ψ α is defined as For example, The quasisymmetric power sums refine the power sum symmetric functions as where the sum ranges over all compositions α whose parts rearrange to λ.This is shown in [BDH + 17, Thm.3.11] and we also give an alternative proof in Section 7.2.
The following expansion of the fundamental basis into quasisymmetric power sums is the main result of this section.
where the sum ranges over all compositions α of n such that S is α-unimodal.
Proof.Expanding both sides of (14) in the monomial basis according to (9) and ( 11) we obtain Comparing coefficients of M β , it suffices to prove that for all compositions β and γ of n, we have where R(β, γ) denotes the set of all compositions α ≤ β such that S γ is α-unimodal.
We give a short alternative proof of this identity in Proposition 3.2 below using the hook-length formula for forests.After multiplying both sides of (15) by n! the claim follows from Theorem 2.5.
Proof.To the pair (α, β) we associate a (labeled) rooted forest on the vertices then add an edge from s i+1 to s i .For example, if α = 2312 and β = 62 then the forest is shown in Figure 2.
The hook length of a vertex in the forest is defined as the size of the subtree rooted at that vertex, see Figure 2. From the definitions in (10) it is straightforward to see that the product of all hook lengths in the forest is π(α, β).Furthermore it is immediate from the definition that the linear extensions of the forest can be identified with Cons(α, β).The hook-length formula for counting linear extensions of forests by D. Knuth [Knu98,Chap. 5.1.4,Ex. 20] Theorem 3.1 immediately implies the following result.
Corollary 3.3.Let X be a quasisymmetric function with Then From Corollary 3.3 and (12) we obtain a new proof of the following result due to C. Athanasiadis for symmetric functions, which has served as inspiration for Theorem 3.1.

Reverse P -partitions
In this section it is shown that the generating function of reverse P -partitions expands positively into quasisymmetric power sums for all posets P .The theory of P -partitions was developed by R. Stanley [Sta72], and has numerous applications in the world of quasisymmetric functions.Let (P, w) be a labeled poset.A function f : P → N + is called reverse (P, w)-partition if it satisfies the following two properties for all x, y ∈ P : (ii) If x < P y and w(x) > w(y) then f (x) < f (y).
Let A r (P, w) be the set of reverse (P, w)-partitions, and denote the generating function of reverse (P, w)-partitions by If (P, w) is naturally labeled, then reverse (P, w)-partitions are just order-preserving maps f : P → N + , which are also called reverse P -partitions.Denote the set of reverse P -partitions by A r (P ) and the corresponding generating function by If instead w is order-reversing then reverse (P, w)-partitions are strict reverse Ppartitions.
The expansion of K P,w into fundamental quasisymmetric functions is a well-known result.
Lemma 4.1 ([Sta01, Cor.7.19.5]).Let (P, w) be a labeled poset with n elements.Then the series K P,w (x) is quasisymmetric and its expansion into fundamental quasisymmetric functions is given by The following theorem is the main result of this section.Theorem 4.2.Let (P, w) be a naturally labeled poset with n elements.Then In particular, the quasisymmetric function K P is Ψ-positive.
Proof.By Lemma 4.1 Expanding the right hand side into quasisymmetric power sums according to Theorem 3.1, we obtain The claim follows directly from Theorem 2.9.
The following is an immediate consequence of Theorem 4.2 and (12).
Corollary 4.3.Let P be a finite set of posets.If the sum is symmetric, then it is p-positive.
We obtain analogous results for the generating functions of strict reverse Ppartitions.
Lemma 4.4 (See also [LMvW13, Chap.3.3.2]).Let (P, w) be a labeled poset with n elements.Then the automorphism ω acts in an order-reversing manner as Proof.This follows from the fact that where In particular, the generating function of strict reverse P -partitions is given by ωK P (x).
Corollary 4.5.Let (P, w) be a labeled poset such that w is order-reversing.Then ωK P,w is Ψ-positive.
Corollary 4.6.Let P be a finite family of labeled posets (P, w), all of which are equipped with an order-reversing map w.If Note that K P,w is not in general Ψ-positive if w is not a natural labeling.Moreover positive linear combinations of quasisymmetric functions K P,w are not in general p-positive whenever they are symmetric.
For example, Corollary 4.6 shows that K P,w is not in general Ψ-positive if w is order-reversing, because the automorphism ω can introduce signs.
Another example is provided by Schur functions s λ , which are shown to be special cases of the functions K P,w in [Sta01,Sec. 7.19].The expansion of Schur functions into power sum symmetric functions is given by the celebrated Murnaghan-Nakayama rule, which is not positive, see [Sta01, Cor.7.17.5].
It is conjectured [Sta72, p. 81] that K P,w is symmetric if and only if it is a skew Schur function.Among these, w is order-preserving only if K P,w is equal to a complete homogeneous symmetric function h λ .It is an interesting question whether the expansion of the quasisymmetric functions K P,w into quasisymmetric power sums can be used to obtain new insights in this regard.

Order-preserving surjections
The purpose of this section is to present a different characterisation of the set L * α (P, w) associated to a naturally labeled poset (P, w).In the process we eschew α-unimodal linear extensions in exchange for order-preserving surjections.This new point of view is then used to compute three examples that are related to matroid quasisymmetric functions, chromatic quasisymmetric functions and Eulerian quasisymmetric functions.Moreover in Theorem 5.4 below we formulate an equivalent version of Theorem 4.2 using order-preserving surjections.
Let P be a poset with n elements.Denote by O(P ) the set of order-preserving surjections f : Furthermore, let O * (P ) denote the set of order-preserving surjections f ∈ O(P ) such that f −1 (i) contains a unique minimal element for all i ∈ [ (α(f ))].That is, for all y, z ∈ P with f (y) = f (z) there exists x ∈ P with x ≤ P y and x ≤ P z and is just the set of linear extensions of P .Compared to linear extensions, order-preserving surjections onto chains have not received much explicit attention.Nevertheless some combinatorial objects can be regarded as order-preserving surjections in disguise.For example, O * (2 m ) (λ), where λ denotes the Young diagram of a partition of n = 2m, is just the set of domino tableaux of shape λ, see [vL00].The set O * (2 m ) (P ) can be taken as the definition of P -domino tableaux [Sta05,Sec. 4].Similarly, ribbon tableaux which appear in the study of Schur functions can be seen as certain order-preserving surjections on Young diagrams.We do not pursue this direction any further in this paper.
Let (P, w) be a naturally labeled poset with n elements, and let α be a composition of n with parts.Given a permutation σ ∈ L * α (P, w) define the map Here we use the notation from Equation (7).
The following proposition is the key result of this section and relates the set L * α (P, w) to order-preserving surjections onto chains.
Proposition 5.1.Let (P, w) be a naturally labeled poset with n elements, and let α be a composition of n.Then the correspondence σ → f σ defines a bijection ϕ : Proof.Let denote the number of parts of α.It is not difficult to see that the map ϕ : Lemma 2.8 implies σ = τ .To see that ϕ is surjective let f ∈ O * α (P ).Define σ ∈ S n as the unique permutation such that σ(B α i ) = w(f −1 (i)) for all i ∈ [ ] and DES(σ) ⊆ S α .That is, the word σ 1 • • • σ n is obtained by first listing the numbers in w(f −1 (1)) in increasing order, then the numbers in w(f −1 (2)) in increasing order, and so on.
(P, σ −1 • w) Note that Corollary 5.3 also follows from Theorem 4.2.Proposition 5.1 even provides a bijection between L * α (P, w) and L * α (P, w ).Another advantage of order-preserving surjections over unimodal linear extensions is that the sets O α (P, w) for α n are disjoint whereas the sets L * α (P, w) intersect.It is therefore much more convenient to work, say, with the set O * (P ) than with its pendant in the world of α-unimodal linear extensions.
The main result of the previous section, Theorem 4.2, has an equivalent formulation in terms of order-preserving surjections.
Theorem 5.4.Let P be poset with n elements.Then Theorem 5.4 yields the intriguing identity We are currently unable to provide such bijections, although we suspect it might not be hopelessly difficult to find them.Note that the first part of the proof of Theorem 2.5 solves Problem 5.5 in the case where P is a chain.A full solution to Problem 5.5 should give an independent proof of Theorems 4.2 and 5.4. 3 We also suspect that guessing (18) without proving Theorem 4.2 first would have been very difficult.
We now compute the numbers |O * α (P )| for three examples.In later sections we use these examples and apply Theorem 5.4 to three families of quasisymmetric functions.The first example is related to the quasisymmetric functions of uniform matroids, see Section 7.8.
Example 5.6 (The complete bipartite graph).Let P be the poset with ground set {x 1 , . . ., x r , y 1 , . . ., y m } and cover relations x i < y j for all i ∈ [r] and j ∈ [m].Thus the Hasse diagram of P is the complete bipartite graph K r,m .
Suppose f ∈ O * (P ).Then f restricts to a bijection f : Moreover, there exists a subset S ⊆ [m] such that f (y j ) = r for all j ∈ S and f restricts to a bijection where k := |S|.We conclude that The next example is connected to Eulerian quasisymmetric functions and chromatic quasisymmetric functions of paths.This is explained in Section 7.9.
Example 5.7 (The path).Let α be a composition of n with parts.We want to compute where P S denotes the poset on {x 1 , . . ., x n } with cover relations x i < x i+1 for i ∈ [n − 1] \ S and x i > x i+1 for i ∈ S. Thus the Hasse diagram of P S viewed as an (undirected) graph is a path.Arranging the vertices x 1 , . . ., x n from left to right, we can think of the Hasse diagram of P S as a word W S ∈ {u, d} n−1 with up-and down-steps, where the down-steps correspond to elements of S.
In this way the poset P in Example 5.2 corresponds to the set S = {3, 5} and the word uududuu.
Let S ⊆ [n − 1] and fix some f ∈ O * α (P S ).For each i ∈ [ ] the subgraph of the Hasse diagram of P S induced by f −1 (i) can be identified with a word W (i) that consists of r i down-steps followed by α i − r i − 1 up-steps.This follows from the fact that f −1 (i) has a unique minimal element.Moreover, reading the values of f from left to right without repetitions we obtain a permutation π.
We claim that the map ψ defined by f → (π, r 1 , . . ., r ) is a bijection ψ : To see this we construct the inverse of ψ.Given (π, r 1 , . . ., r ) first form words W (i) consisting of r i down-steps followed by α i − r i − 1 up-steps.We recover S from the identity where a i = d if i ∈ DES(π) and a i = u otherwise.Once α, S and π are known, it is easy to recover f .
From |S| = des(π) + i=1 r i it follows that where A k (q) := σ∈S k q des(σ) is the Eulerian polynomial and [a] q := 1−q a 1−q is the commonly used q-integer. 2 Our final example is related to chromatic quasisymmetric functions of cycles and cycle Eulerian quasisymmetric functions, see Section 7.9.The same computation was previously done by B. Ellzey with different notation.For example, the following poset corresponds to the set S = {2, 3, 4, 6, 7} and the word udddudd.
Let S ⊆ [n] with 0 < |S| < n.We have to distinguish two cases.
Case 1.First assume that ≥ 2. Fix some f ∈ O * α (P S ).For each i ∈ [ ] the subgraph of the Hasse diagram of P S induced by f −1 (i) is a path and can be identified with a word W (i) that consists of r i down-steps followed by α i − r i − 1 up-steps.Again this follows from the fact that f −1 (i) has a unique minimal element.Reading the values of f from left to right without repetitions we obtain a long cycle (π 1 , . . ., π −1 , ) ∈ S which we identify with the permutation π 1 • • • π −1 ∈ S −1 .Moreover, let k be the position of the minimal element of f −1 (1).
We claim that the map ψ defined by f → (k, π, r 1 , . . ., r ) is a bijection ψ : To see this we construct the inverse of ψ in a similar fashion as in Example 5.7.Given (k, π, r 1 , . . ., r ) first form words W (i) consisting of r i down-steps followed by α i − r i − 1 up-steps.Then W S is a cyclic shift of the word where a i = d if i ∈ DES(π) and a i = u otherwise.The number k contains precisely the information needed to recover S. As before, f can be obtained once α, S and π are known.
For example, suppose we are given the data (k, π, r 1 , r 2 , r 3 ) = (5, 21, 2, 1, 0).We first form the posets corresponding to the words W (i). Since the minimal element of f −1 (1) is currently in sixth position, we have to cyclically shift until the position is k = 5 to obtain W S .
From |S| = 1 + des(π) + i=1 r i it follows that where A k (q) and [a] q are defined as in Example 5.7.
Note that O * (n) (P S ) contains a unique element if P S has a unique minimal element, and O * (n) (P S ) is empty otherwise.It is not difficult to see that

Reverse P -partitions with forced equalities
In this section we consider the generating function of reverse P -partitions that assign equal colors to certain given elements.In Theorem 6.2 below we show that these functions are Ψ-positive, thus extending Theorem 4.2.We hope this will enable (or simplify) future applications of our results to families of quasisymmetric functions that are not covered by Section 4. A simple example of such a family are the power sum symmetric functions p λ , which we discuss in Section 7.2.Reverse P -partitions with forced equalities have not received much attention to date.The results of this section might be an indication that they are investigating further.
A partitioned poset (P, w, E) is a naturally labeled poset (P, w) endowed with an equivalence relation E on P .Given a partitioned poset (P, w, E) with n elements, let A r (P, E) := f ∈ A r (P ) : f (x) = f (y) for all x, y ∈ P with x E ∼ y and define the generating function This is a homogeneous quasisymmetric function of degree n.For example, if all equivalence classes of E are singletons then K P,E = K P .On the other hand if x E ∼ y for all x, y ∈ P , then K P,E equals the power sum symmetric function p n .
In the above definition the equivalence relation E is completely arbitrary.We now define a very special kind of equivalence relation.
Let P be a poset.An equivalence relation E on P is a chain congruence on P if the following two conditions are satisfied: (i) The equivalence class [x] E is a chain in P for all x ∈ P .(ii) For all x, y ∈ P with x < P y and x E y we have max [x] E < P min [y] E .
In order to prove Ψ-positivity of the functions K P,E it suffices to consider chain congruences.This is the content of the next lemma.Lemma 6.1.Let (P, w, E) be a partitioned poset.Then there exists a partitioned poset (P , w , E ) such that K P,E (x) = K P ,E (x) , the poset P is obtained from P by adding order relations, E is obtained from E by joining equivalence classes, and E is a chain congruence on P .
Proof.Define a relation E on P as follows.For x, y ∈ P let x E ∼ y if and only if the following two (symmetric) conditions are satisfied: (i) There exist k ∈ N and x 1 , y 1 , . . ., x k , y k ∈ P such that x = x 1 , y = y k , x i ≤ P y i for all i ∈ [k], and and See Figure 3 for an example.
Clearly E is an equivalence relation and x E ∼ y implies x E ∼ y.We claim that K P,E = K P,E .To see this note that A r (P, E ) ⊆ A r (P, E).Thus let f ∈ A r (P, E) and x, y Next define a relation ≤ P on P as follows.For all x, y ∈ P let x ≤ P y if and only if one of the following two mutually exclusive conditions is satisfied: (i) We have x E ∼ y and w(x) ≤ w(y).
(ii) We have x E y and there exist k ∈ N and x 1 , y 1 , . . ., x k , y k ∈ P such that Then P is a poset with the same ground set as P , and x ≤ P y implies x ≤ P y for all x, y ∈ P .Let w be an arbitrary natural labeling of P .
We claim that K P ,E = K P,E .To see this note that A r (P , E ) ⊆ A r (P, E ).Thus let f ∈ A r (P, E ) and x, y ∈ P with x ≤ P y.The definition of P implies f (x) ≤ f (y) and therefore f ∈ A r (P , E ).
It remains to show that E is a chain congruence on P .Clearly [x] E is a chain by definition of P for all x ∈ P .Secondly suppose x < P y and x E y for some x, y ∈ P .Then Hence max [x] E ≤ P min [y] E and the proof is complete.
Let (P, w, E) be a partitioned poset.If we are only interested in K P,E , then by Lemma 6.1 we may assume that E is a chain congruence on P .If this is the case we can form the quotient poset P/E, that is, the partial order on the set of equivalence classes {[x] E : x ∈ P } defined by [x] E ≤ [y] E if and only if x ≤ P y.Then This leads to a second, equivalent, way of thinking about partitioned posets.
A weighted poset (P, w, d) consists of a naturally labeled poset (P, w) and a vector d = (d x ) x∈P of positive integers -a weight on P .Define the weighted generating function This is a homogeneous quasisymmetric function of degree |d|.For example, if P is a poset with one element then K d P (x) is equal to the power sum symmetric function p |d| (x).
The following theorem is the main result of this section and generalizes Theorem 4.2.Theorem 6.2.Let (P, w) be a naturally labeled poset with n elements, and let E be a chain congruence on P .Then where L * α (P, w, E) denotes the set of permutations σ ∈ L * α (P, w) such that for each x ∈ P there exists i ∈ [ (α)] with [x] E ⊆ P α i (σ).Note that min P α i (σ) is well-defined for all σ ∈ L * α (P, w).Equivalently, where O * α (P, E) denotes the set of order-preserving surjections f ∈ O * α (P ) that satisfy f (x) = f (y) for all x, y ∈ P with x E ∼ y.Note that min f −1 (i) is well-defined for all f ∈ O * α (P ).In particular, K P,E is Ψ-positive.Theorem 6.2 has an equivalent formulation in terms of weighted posets.Theorem 6.3.Let (P, w, d) be a weighted poset with n elements.Then where β = β(d, α, σ) is the composition of |d| defined by Equivalently, where β = β(d, α, f ) is the composition of |d| defined by In particular, Proof of equivalence of the statements in Theorems 6.2 and 6.3.To see that ( 19) and ( 20) are equivalent, note that the bijection σ → f σ from Proposition 5.1 restricts to a bijection from L * α (P, w, E) to O * α (P, E).The claimed equivalence follows from P α i (σ) = f −1 σ (i).The equivalence of ( 21) and ( 22) follows from Proposition 5.1 as above.
To see that Theorem 6.3 implies Theorem 6.2 let E be a chain congruence on a finite poset P , and let P/E be the associated quotient poset.

We prove the claim by induction on n − c(E).
Clearly the case c(E) = n reduces to Theorem 4.2, so we may assume that there exists C ∈ C (E) with |C| ≥ 2, and that the claim holds for all chain congruences on P with more equivalence classes.Let C = {x 0 , . . ., x k } for some k ∈ [n − 1], so that x 0 = min C and Define two new equivalence relations on P by Moreover define a new naturally labeled poset (P , w) by removing all order relations x i < x k for i ∈ {0, . . ., k − 1}, see Figure 4.
Note that E and E are chain congruences on P , and that E is a chain congruence on P .Thus K P,E , K P,E and K P ,E satisfy (20) by the induction hypothesis.
The function K P,E satisfies the recursion This identity follows directly from the definition of K P,E and is analogous to the fact that Now fix a composition α of n with parts and a surjective map f : P → [ ] of type α.We need to show that the contribution of f predicted by (20) is the same on both sides of (24).
First assume that f ∈ O * α (P, E).Then there exists , E ) and that f ∈ O α (P ) is an order-preserving surjection of type α.
On the other hand, if min f −1 (i) < P x 0 then f ∈ O * α (P , E ).The contribution of f on both sides of (24) is To finish the proof we need to show that the contributions of surjective maps f : P → [ ] of type α that do not lie in O * α (P, E) on the right hand side of (24) cancel.

Let
O + α (P , E ) := {f ∈ O * α (P , E ) : f (x 0 ) < f (x k )} , and similarly ), and the terms corresponding to f cancel., E ), and that the terms coming from f and f cancel.
Hence we are in one of the cases above, and the proof is complete.
An immediate consequence of Lemma 6.1 and Theorem 6.2 is the following result.
Note that Theorem 6.2 does not apply unless E is a chain congruence.Furthermore, Theorem 6.3 requires that the weight d is indeed a vector of positive integers.For example, K d P given by the poset x < y < z with weights d x = 1, d y = 0 and d z = 2 is not Ψ-positive.

Applications
In this section we apply Theorems 5.4 and 6.3 to derive the expansions of various families of quasisymmetric functions into quasisymmetric power sums.

Complete homogeneous symmetric functions. As a warm up we now derive the expansion of the complete homogeneous symmetric function h λ into power sum symmetric functions.
Let P λ be the poset consisting of disjoint chains of lengths λ i .Then h λ = K P λ and by Theorem 5.4.
Another interpretation of these coefficients is given by so called ordered µ-bricks, see [ER91].An ordered µ-brick tabloid of shape λ is a Young diagram of shape λ filled with labeled bricks of sizes given by µ.The bricks are placed in the diagram such that the bricks in each row are sorted with increasing label.Let OB µλ be the set of such ordered µ-brick tabloids of shape λ.
It is easy to see that O * µ (P λ ) is in bijection with OB µλ .For instance, let λ = 532 and µ = 322111.Then O * µ (P λ ) contains the following order-preserving surjection, which is in natural correspondence with the shown ordered brick tabloid.7.2.Power sum symmetric functions.We have already stated in (12) the expansion of power sum symmetric functions p λ into quasisymmetric power sums Ψ α .We now give an independent proof of this result using the tools we have developed.
7.3.Schur functions.While Schur functions are not p-positive, as is evident from the famous Murnaghan-Nakayama rule, we still obtain an expansion of Schur functions into quasisymmetric power sums from the results in Section 3. The well-known formula below expresses the Schur functions in the fundamental quasisymmetric functions (see for example [Sta01, Thm.7.19.7]): The set SYT(λ) is the set of standard Young tableaux of shape λ n, and the descent set of such a standard Young tableau is the set of entries i such that i + 1 appears in a row with a higher index.The following result due to Y. Roichman is an immediate consequence of (25) and Corollary 3.4.
Theorem 7.1 ([Roi97, Thm.4]).Let λ be a partition of n.Then the expansion of the Schur function s λ into power sum symmetric functions is given by Example 7.2.Let λ = (3, 3).Then SYT(λ) is the following set of standard Young tableaux, where the descents have been marked bold.If µ = (2, 2, 2) then S µ = {2, 4}.We can check that the descent sets of all five tableaux are µ-unimodal, and contribute the signs −1, +1, −1, −1 and −1 respectively.Furthermore, U µ is the set of all subsets of [5].The formula in (26) then gives In contrast, the Murnaghan-Nakayama rule -given as a sum over so called rimhook tableaux -is cancellation-free for this choice of λ and µ, and is given as a sum over exactly three rim-hook tableaux. .This definition was later refined by J. Shareshian and M. Wachs in [SW16], where a q-parameter was introduced.
Definition 7.3.Let G be a directed graph (no loops, but multiple edges are allowed) on the vertex set [n].A coloring of G is an assignment of colors in N + to the vertices.A coloring is proper if vertices connected by an edge are assigned different colors.
An ascent4 of a coloring κ is a directed edge (i, j) of G such that κ(i) < κ(j).The number of ascents of a coloring is denoted asc(κ).
The chromatic quasisymmetric function of G is defined in [SW16] 5 as κ) . 2 When q = 1 we obtain the chromatic symmetric function X G (x) in [Sta95].The function X G (x) is a symmetric function, and does not depend on the orientation of G. R. Stanley proves that ωX G (x) is p-positive for any (undirected) graph G.
For some choices of directed graphs G, X G (x; q) is a symmetric function.A class of such graphs is characterized by B. Ellzey in [Ell16, Thm.5.5].In particular, if G is the incomparability graph of a 3 + 1 and 2 + 2-avoiding poset (together with a certain associated orientation), then the function X G (x; q) is symmetric -these graphs are referred to as unit-interval graphs.
B. Ellzey also proves that X G (x; q) has the property that ωX G (x; q) expands positively in the power-sum symmetric functions whenever it is symmetric.The only part of her proof that requires the restriction to symmetric functions is the application of Corollary 3.4.We can now prove the following generalization of the main result in [Ell16, Thm.4.1]: Theorem 7.4.Let G be a directed graph and consider the expansion into quasisymmetric power sums Then c G α (q) ∈ N[q] for all compositions α.
Proof.Let AO(G) denote the set of acyclic orientations of the graph G viewed as an undirected graph.For an orientation θ, we let asc(θ) be the number of edges oriented in the same direction as in G.It is straightforward to prove (see [AP17,Ell16]) that where P (θ) is the poset obtained from θ by taking the transitive closure of the directed edges, and w = w(θ) is order-reversing.The statement now follows from Corollary 4.5.
Note that different types of combinatorial interpretations for the coefficients c G α (q) in (7.4) are known in certain special cases, see [Ath15,Ell16].Our approach yields a combinatorial interpretation in the more general setting, which is similar to [Ell16].
It was conjectured in [SW16, Conj.7.6] that the coefficients c G α (q) are unimodal for unit-interval graphs.This conjecture is still open.However, this conjecture does not extend to the general quasisymmetric setting.
Example 7.5.Consider the following directed graphs G and H.
Note that the coefficient of Ψ 121 is not unimodal.
For the graph H we have where the coefficient of Ψ 131 is not unimodal.2 7.5.k-balanced chromatic quasisymmetric functions.B. Humpert introduces another quasisymmetric generalization of chromatic symmetric functions in [Hum11].
Definition 7.6.Let G be an oriented graph (no loops or multiple edges) on the vertex set [n] and let k ∈ N + .An orientation θ of G is said to be k-balanced if for every undirected cycle in G, walking along the cycle one traverses at least k edges forward, and at least k edges backwards.Thus an orientation is acyclic if and only if it is 1-balanced.
A proper coloring κ of G induces an acyclic orientation θ(κ) of G by orienting edges towards the vertex with larger color.
The k-balanced chromatic quasisymmetric function ([Hum11] defined this only for q = 1) is defined as 2 Note that for k = 1 we recover the quasisymmetric function in Definition 7.3 as X 1 G (x; q) = X G (x; q).Proposition 7.7 ([Hum11, Thm.3.4]).The k-balanced chromatic quasisymmetric function of an oriented graph G has the expansion where the sum taken is over all k-balanced orientations of G, P (θ) is the transitive closure of the directed edges and w = w(θ) is order-reversing.
From Proposition 7.7 and Corollary 4.5 we obtain the following consequence.

Corollary 7.8. Let G be any oriented graph and consider the expansion into quasisymmetric power sums
Then c G α (q) ∈ N[q] for all compositions α.
7.6.LLT polynomials.The LLT polynomials were introduced by A. Lascoux, B. Leclerc and J.-Y.Thibon in [LLT97] using ribbon tableaux.The LLT polynomials can be seen as q-deformations of products of Schur functions and there are several open problems regarding LLT polynomials.A different combinatorial model for the LLT polynomials was considered in [HHL05], where each k-tuple of skew shapes index an LLT polynomial.When each such skew shape is a skew Young diagram with a single box, we say that the LLT polynomial is unicellular.The unicellular LLT polynomials have a central role in the work of E. Carlsson and A. Mellit [CM17], in which they introduced a combinatorial model for the unicellular LLT polynomials using Dyck paths.In [AP17] this Dyck path model was extended to certain directed graphs.
By modifying the definition of the chromatic symmetric functions slightly, we recover the unicellular LLT polynomials considered in [AP17]: Definition 7.9.Let G be a directed graph (no loops, but multiple edges are allowed) on the vertex set [n].The unicellular graph LLT polynomial is defined as Note that we now sum over all colorings. 2 The G G (x; q) are in general only quasisymmetric, but for certain choices of G (the same choices as for the chromatic quasisymmetric functions) they turn out to be symmetric and contain the family of unicellular LLT polynomials, see [AP17].
It was observed in [AP17,HW17] that ωG G (x; q + 1) is p-positive whenever G G (x; q) is a unicellular LLT polynomial.We can now give a proof of the following much stronger statement.
Theorem 7.10.Let G be a directed graph and consider the expansion into quasisymmetric power sums Then c G α (q) ∈ N[q] for all compositions α.
Proof.Let O(G) denote the set of orientations of the graph G viewed as an undirected graph.For θ ∈ O(G) we let asc(θ) be the number of edges oriented in the same direction as in G. Similar as in the proof of Theorem 7.4, we have q asc(θ) K P (θ),w (x) where P (θ) is the transitive closure of only the edges of θ oriented in the same manner as in G and w = w(θ) is order-reversing.Note that we let K P (θ),w (x) := 0 when P (θ) has a cycle -this can only happen if G has a directed cycle.
Again the result follows from Corollary 4.5.
We can enlarge the family of unicellular graph LLT polynomials.
Definition 7.11.Let G be a directed graph on the vertex set [n] and let S be a subset of the edges of G.The vertical strip graph LLT polynomial is defined as where we sum over all colorings such that κ(i) < κ(j) whenever (i, j) is a (directed) edge in S. 2 The name "vertical strip graph LLT polynomials" is motivated as follows.For some choices of G and S we recover the family of LLT polynomials that are, in the model introduced in [HHL05], indexed by k-tuples of vertical strips.Vertical strip LLT polynomials occur naturally in the study of the delta operator and diagonal harmonics.The family of vertical strip LLT polynomials contains (a version of) modified Hall-Littlewood polynomials.See [AP17] for an explicit construction of the correspondence between the above model and the model in [HHL05].
Theorem 7.12.Let G be a directed graph, S a subset of the edges of G, and let be the expansion into quasisymmetric power sums.Then c G α (q) ∈ N[q] for all compositions α.
Proof.The same technique as above (also in [AP17]) shows that G G,S (x; q + 1) = θ∈O S (G) q asc(θ)−|S| K P (θ),w (x) where O S (G) is now the subset of orientations of G such that edges in S are oriented as in G.
A special case of Theorem 7.12 was proved in [AP17].
It is conjectured by P. Alexandersson and G. Panova in [AP17] that the coefficients c G α (q) in Theorem 7.10 are unimodal whenever G is a unit interval graph.Computer experiments suggests that this conjecture extends to the more general setting in Theorem 7.10.

Conjecture 7.13. Let G be an oriented graph (no loops or multiple edges). Then the coefficients c
are unimodal for all compositions α.
Conjecture 7.13 has been verified for all oriented graphs with six or fewer vertices. 6In contrast, we note that the coefficients c G,S α (q) in (28) are not unimodal in general.For example, It is possible to refine Theorems 7.4, 7.10 and 7.12 by assigning a different q-weight to each edge of G, so that for a coloring κ we let q asc(κ) := (i,j)∈E(G) κ(i)<κ(j) q i,j .
The resulting functions are again quasisymmetric and the analogues of the above theorems can be proved in the same manner.We leave out the details.
6 See [Slo16, A001174] for the number of such graphs.
where the sum ranges over all vertex colorings of G and m(κ) denotes the number of monochromatic edges -edges {i, j} such that κ(i) = κ(j).It is evident that this is a symmetric function and it is straightforward to prove (see [Sta98]) that Tutte G (x; q) = S⊆E(G) where the sum ranges over all subsets of the edges of G, and λ(S) is the partition whose parts are the sizes of the connected components of the subgraph of G spanned by the edges in S.
where asc(κ) and inv(κ) are defined as Notice that the chromatic quasisymmetric function can be obtained as X G (x; q) = [z n ]B G (x; qz, z), and that the unicellular graph LLT polynomials can be obtained as G G (x; q) = B G (x; q, 0).Furthermore, for any directed graph G, the Tutte polynomials satisfies the relationship where G denotes the undirected version of G.
Theorem 7.15.Let G be a directed graph and consider the expansion Proof.Let E(G) be the set of directed edges of G. Then we have that where P (A, I) is the transitive closure of the directed edges and w is an order-reversing labeling of P (A, I).Here we let K P (A,I),w := 0 if some edges in (31) form a cycle.By Corollary 4.5, the statement follows.
7.8.Matroid quasisymmetric functions.In 2009 L. Billera, N. Jia and V. Reiner introduced a quasisymmetric function associated to matroids as a new matroid invariant, see [BJR09].The definition is as follows: Definition 7.16.Let M be a matroid with ground set E and bases B(M ).A map f : E → N + is said to be M -generic if the sum f (B) := e∈B f (e) is minimized by a unique B ∈ B(M ).An M -generic function f must also be injective.
The matroid quasisymmetric function is then defined as In [BJR09] it is proved that F (M, x) is indeed a quasisymmetric function.Theorem 7.17 ([BJR09, Thm.5.2]).Let M = (E, B(M )) be a matroid.Then where w is any order-reversing labeling of P B .
Using Corollary 4.5 we get the following corollary.
We compute the Ψ-expansion of the matroid quasisymmetric function F (M, x) explicitly in the case where M is the uniform matroid.
Example 7.19 (Uniform matroid).The uniform matroid U = U r n has ground set E = [n] and every r-element subset of E constitutes a basis.That is, B(U ) = [n]  r .In this case the poset P B is given by e ≺ e for all e ∈ B and e ∈ B * .In particular all posets P B for B ∈ B(U ) are isomorphic.The Hasse diagram of P B is the complete bipartite graph K r,m , where m = n − r.
Fix a basis B ∈ B(U ) and a natural labeling w of P B .It follows from Theorem 7.17, Theorem 5.4 and Example 5.6 that 7.9.Eulerian quasisymmetric functions.The aim of this section is to explain how the tools developed in this paper can be used to prove known p-expansions of the Eulerian quasisymmetric functions and the cycle Eulerian quasisymmetric functions of J. Shareshian and M. Wachs.
The Eulerian quasisymmetric functions are defined as In [SW10, Thm.1.2] Eulerian quasisymmetric functions are shown to have the following generating function that specializes to (32).
Here H(x; z) := n≥0 h n (x)z n denotes the generating function of complete homogeneous symmetric functions.
We present an alternative proof of Proposition 7.20 using the theory of orderpreserving surjections and an interpretation of Eulerian quasisymmetric functions as generating functions of banners, which was obtained in [SW10, Sec.3.2].Note that this also offers a different route to proving that Q n,j is symmetric and satisfies (34).
Let X and X denote disjoint copies of the positive integers, that is, in the alphabet X ∪ X such that the following three conditions are satisfied: Let B n,j denote the set of banners of length n that contain exactly j barred letters.Given a banner b ∈ B n,j define its weight as x b := x |b1| where G denotes the directed path of length n − 1.
Using a few tricks one can also deal with the more challenging cycle Eulerian quasisymmetric functions Q (n),j , which are defined as The Eulerian quasisymmetric functions Q n,j can be expressed in terms of cycle Eulerian quasisymmetric functions Q (n),j (and vice versa) via plethysm.J. Shareshian and M. Wachs conjectured and later proved together with B. Sagan the following expansion into power sum symmetric functions.
Note that our statement of Theorem 7.21 differs slightly from the given references.See [SSW11,Lem. 4.2] for a proof that the statements are equivalent.
In [SSW11] Theorem 7.21 is derived from Proposition 7.20 by the use of plethystic calculus and the manipulation of formal power series.We present a proof based on the theory of order-preserving surjections, Möbius inversion and an interpretation of the cycle Eulerian quasisymmetric functions as the generating functions of primitive necklaces due to J. Shareshian and M. Wachs.The application of Möbius inversion to problems related to primitive necklaces is very classical and dates back (at least) to [MR83].In this way the Möbius function and the power q d on the right hand side of (38) appear naturally.
Let X, X and | • | be as in (35).A bicolored necklace of length n is a circular word o 1 • • • o n in the alphabet X ∪ X that satisfies the three conditions below.Circular means that we do not distinguish between o The quasisymmetric functions Q (n),j are not positive linear combinations of reverse P -partition enumerators K P for some simple set of posets P .Instead, let B n,j be the set of words b 1 • • • b n in the alphabet X ∪ X that satisfy the following three properties: where x k denotes the variables x k 1 , x k 2 , . . . .Contrary to the cycle Eulerian quasisymmetric functions Q (n),j , the generating functions F n,j are immediately related to reverse P -partitions and chromatic quasisymmetric functions of cycles.More precisely, where P S is defined as in Example 5.8, and G denotes the directed cycle of length n.
Then X is symmetric if and only if c α = c β for all compositions α and β such that β can be obtained by permuting the parts of α.
Proposition 8.1 offers a method for proving (or disproving) that a given quasisymmetric function is symmetric.As mentioned at the end of Section 4, it would be particularly interesting if this idea was applicable to the generating functions K P,w .
Furthermore by Proposition 8.1 any symmetric function for which the expansion into quasisymmetric power sums is known, immediately gives rise to a set of symmetries on its coefficients c α .These symmetries might not at all be obvious from a purely combinatorial point of view.For example, consider the expansion of Schur functions into quasisymmetric power sums given in Section 7.3.It follows from the symmetry of Schur functions that the sum T ∈SYT(λ) DES(T )∈Uα (−1) DES(T )\Sα is invariant under the permutation of the parts of α.The authors are unaware of a proof of this fact that does not appeal to the theory of symmetric functions.We expect that many potentially interesting combinatorial problems can be obtained in similar fashion.
Lastly, if a combinatorial statistic c on compositions appears to satisfy the symmetry properties of Proposition 8.1, then it might be worth a try to investigate the quasisymmetric function Proving that X n is symmetric, for example by deriving its expansion into the fundamental or monomial bases, will also prove the symmetry of c.

Schur-positivity and h-positivity.
There are open problems regarding a combinatorial proof of the Schur-positivity of LLT polynomials [HHL05], as well as proving e-positivity of chromatic symmetric functions [SS93].
Since both families of polynomials are related to the K P (x), it is natural to ask if every symmetric positive linear combination of such functions is h-positive (or weaker, Schur-positive).This is not the case.A computer search gave us the following counterexamples: For the above posets where the sum is over all µ such that µ/λ is a ribbon of size r.A natural future direction is then to seek quasisymmetric refinements or analogues of the Murnaghan-Nakayama rule.Let {X α } be any of the above families of quaissymmetric functions indexed by partitions.The problem is then to find a rule that gives the coefficients χ γ αβ in the expansion Possible research in this direction is also discussed in [BDH + 17, Sec.7.1].
8.4.Poset invariants.Whenever a class of combinatorial objects has a nontrivial isomorphism problem (such as posets, graphs or knots) it immediately becomes an interesting task to find invariants that might be used to distinguish such objects.
In [MW14] P. McNamara and R. Ward ask for a necessary and sufficient condition that two labeled posets (P, w) and (Q, w ) have the same P -partition generating function, that is, K P,w = K Q,w .
For naturally labeled posets Theorem 5.4 yields that K P = K Q if and only if |O * α (P )| = |O * α (Q)| for all compositions α.That is, the numbers of certain order-preserving surjections onto chains agree.Note that this includes (in the case of naturally labeled posets) the observation [MW14, Prop.3.2] that K P = K Q implies |L(P, w)| = |L(Q, w )|.The judgment whether our answer is more useful than the trivial answer, "K P = K Q if and only if the multisets of descent sets {DES(σ) : σ ∈ L(P, w)} and {DES(σ) : σ ∈ L(Q, w )} agree", is left to the reader.P. McNamara and R. Ward pose several other problems in this direction, many of which where recently solved for naturally labeled posets by R. Liu and M. Weselcouch [LW18].It could be worth investigating whether the Ψ-expansion of K P has applications in this regard.
Another open question that was first raised by R. Stanley in [Sta95, p. 170] is whether the chromatic symmetric function X G (x) defined in Section 7.4 distinguish trees.This was investigated, for instance, by J. Martin, M. Morin and J. Wagner in [MMW08].
Let T be a rooted tree.We interpret T as a poset by declaring the edges to be cover relations, and the root to be the unique minimal element.It was recently shown by T. Hasebe and S. Tsujie that the generating function of reverse P -partitions distinguishes rooted trees [HT17].The following is a straightforward consequence of this result.
Proposition 8.2.The chromatic quasisymmetric function X G (x; q) distinguishes rooted trees with all edges directed away from the root.
Proof.Let T be a rooted tree on n vertices.It follows from the proof of Theorem 7.4 that the coefficient of q n−1 in X T (x; q) is just K T,w , where T is viewed as a poset as above and w is an order-reversing labeling.Thus the claim is a consequence of [HT17, Thm.1.3].
It is an open problem whether X G (x; q) distinguishes all oriented trees.
Let P and Q be (disjoint) posets.Let P + Q denote the direct sum of P and Q, that is, the partial order on P Q defined by x < y if x < P y or x < Q y.Let P ⊕ Q denote the ordinal sum of P and Q, that is, the partial order on P Q defined by x < y if (i) x < P y, or (ii) x < Q y, or (iii) x ∈ P and y ∈ Q. Clearly all rooted trees can be obtained by successively taking direct sums and adding a minimal element, that is, forming the poset { 0} ⊕ P .A poset is called series parallel if it can be built from sigletons using only the two operations direct and ordinal sum.
A very natural question raised by T. Hasebe and S. Tsujie is whether K P distinguishes series parallel posets.The ideas in this paper might offer a new angle to attack this problem, since it is not too difficult to compute the expansions of K P +Q respectively K P ⊕Q into quasisymmetric power sums recursively using Theorem 5.4.
Another possible question is whether other constructions on posets have a simple interpretation in the basis of quasisymmetric power sums.8.5.Type B n analogues.There is a type B analogue of Theorem 7.1 given in [AAER17].This setting uses Schur functions, power sum symmetric functions and fundamental quasisymmetric functions in two sets of variables x and y.It is natural to ask if this result extends to an analogue of Theorem 3.1 that uses some kind of quasisymmetric power sums in two sets of variables.

Figure 2 .
Figure 2. Left: The forest associated with the compositions α = 2312 and β = 62.Right: The hook lengths of the vertices.
is a composition of n = |P | with k parts.Denote by O α (P ) the set of order-preserving surjections of P with type α.

2
An immediate consequence of Proposition 5.1 is the following.Corollary 5.3.Let (P, w) be a naturally labeled poset with n elements, and let α be a composition of n.Then the cardinality |L * α (P, w)| is independent of w.That is, if w : P → [n] is another natural labeling, then |L * α (P, w)| = |L * α (P, w )|.
Example 5.8 (The cycle,[Ell16, Thm.4.4]).Let α be a composition of n with parts.We want to computeS⊆[n] 0<|S|<n q |S| |O * α (P S )| ,where P S denotes the poset on {x 1 , . . ., x n } with cover relationsx i < x i+1 for i ∈ [n] \ S and x i > x i+1 for i ∈ S.Here indices are to be understood modulo n, thus the Hasse diagram of P S viewed as an (undirected) graph is a cycle.Arranging the vertices x 1 , . . ., x n from left to right, we can think of the Hasse diagram of P S as a word W S ∈ {u, d} n , where the last letter determines the relation between x 1 and x n .Note that we have to exclude the cases S = ∅ and S = [n] since they do not give rise to partial orders.

3
Next join the words W (i) according to π (with W (3) in last position).

Figure 3 .E∼ v 4 and v 2 E∼ v 5 and v 3 E∼
Figure 3.A poset P with an equivalence relation E given by v 1 E ∼ v 4 and Every order-preserving surjection f ∈ O * β (P, E) yields an order-preserving surjection f ∈ O * α (P/E) where the type β of f is related to the type α of f by (23).Indeed, this correspondence defines a bijection from O * β (P, E) to O * α (P/E).Define a weight d on P/E by assigning to each equivalence class [x] E its cardinality.Then the formula (22) for K d P/E implies (20).Conversely, let (P, w, d) be a weighted poset.Define a poset Q by replacing each element x ∈ P by a chain C x with d x elements.That is, Q = x∈P C x and for y 1 ∈ C x1 and y 2 ∈ C x2 , where x 1 = x 2 , we have y 1 ≤ Q y 2 if and only if x 1 ≤ P x 2 .The poset Q comes equipped with a chain congruence E, namely y 1 E ∼ y 2 if and only if y 1 , y 2 ∈ C x for some element x ∈ P .Given an order-preserving surjection f ∈ O * α (P ) define an order-preserving surjection f∈ O * β (Q, E) by f (y) = f (x) if y ∈ C x .This defines a bijection from O * α (P ) to O * β (Q, E) where α and β are related by (23).Hence the formula (20) for K Q,E implies (22), and Theorem 6.3 follows from Theorem 6.2.Proof of Theorem 6.2 (20).Given an equivalence relation R let C (R) denote the set of its equivalence classes, and let c(R) = |C (R)| denote the number of equivalence classes.

Figure 4 .
Figure 4.The four partitioned posets from the proof of Theorem 6.2.

J
. Awan and O. Bernardi [AB16] define a quasisymmetric generalization of the Tutte polynomial, which they call the B-polynomial.Definition 7.14.Let G be a directed graph on the vertices [n].Let B G (x; y, z) := Let M = (E, B(M )) be a matroid.Given a basis B ∈ B(M ) let B * := E \ B and define the poset P B on the vertex set E = B B * such that e ≺ e if and only if e ∈ B and (B \ {e}) ∪ {e } is in B(M ).
where indices are viewed modulo n. (ii) If b i ∈ X then |b i | ≥ |b i+1 | for all i ∈ [n], where indices are viewed modulo n. (iii) The word b contains exactly j barred letters.Denote the generating function of B n,j by F n,j (x) := b∈B n,j x b .Note that every word b 1 • • • b n can be written uniquely as (a 1 • • • a d ) n/d where d divides n and a 1 • • • a d is a primitive word of length d.Moreover each primitive circular word o 1 • • • o d gives rise to d distinct primitive words.Therefore the quasisymmetric functions F n,j are related to the cycle Eulerian quasisymmetric functions by n−1 j=0
For S ⊆ [n − 1] let B n,S denote the set of banners b such that b i ∈ X if and only if i ∈ S. Then