A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials $\mathrm{LLT}_P(x;q)$ as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if $q$ is replaced by $q+1$, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

LLT polynomials form a large class of symmetric functions that can be viewed as q-deformations of products of (skew) Schur functions. They appear in many different contexts in representation theory and algebraic combinatorics. In this paper we are concerned with a particular subclass of LLT polynomials, namely the vertical-strip LLT polynomials. They contain information about the equivariant cohomology ring of regular semisimple Hessenberg varieties, and are integral to the study of diagonal harmonics and the proof of the Shuffle Theorem.
Vertical-strip LLT polynomials can be indexed by certain lattice paths called Schröder paths. We view the vertical-strip LLT polynomial G P (x; q) associate to a Schröder path P of size n as the generating function of certain colorings of a unit-interval graph Γ P on n vertices. The main result of this paper is an explicit expansion of G P (x; q) in the basis of elementary symmetric functions as a weighted sum over orientations of Γ P . This expansion is given by G P (x; q) = κ∈C(P ) q asc(κ) x κ(1) x κ(2) · · · x κ(n) = θ∈O(P ) (q − 1) asc(θ) e λ(θ) (x), where C(P ) is a set of colorings, O(P ) is a set of orientations, and asc denotes certain combinatorial statistics on these objects. The expression λ(θ) is an integer partition determined by the orientation θ. In particular, the above identity proves that G P (x; q + 1) is e-positive. This resolves several open conjecture stated in [AP18,Ale20,GHQR19]. Algebraically the e-positivity phenomenon implies that the symmetric function in question is -up to a twist by the sign representationthe Frobenius character of a permutation representation of the symmetric group in which each point stabilizer is a Young subgroup. Our approach to vertical-strip LLT polynomials highlights their similarities with chromatic quasisymmetric functions. The expansion in (1) is of particular interest because it serves as an analog of the Shareshian-Wachs conjecture regarding the e-positivity of chromatic quasisymmetric functions.
We now briefly outline the contents of this paper. Section 2 contains background on LLT polynomials, and all definitions necessary to state our main results. In Theorem 2.1 and Theorem 2.5 we provide sets of relations which uniquely determine the vertical-strip LLT polynomials. Theorem 2.9 states that the right-hand side in (1) satisfies the same set of relations, which implies equality in (1). Section 3 treats linear relations among LLT polynomials and contains the proof of Theorem 2.1. Moreover we give a new characterisation of the (smaller) class of unicellular LLT polynomials in Theorem 3.5. Section 4 contains bijective results on colorings and the proof of Theorem 2.5. Section 5 deals with the combinatorics of orientations, culminating in a proof of Theorem 2.9.
In Section 6 we give an overview of related areas and open problems. We only mention a few here. In Corollary 6.2 we state a new signed Schur expansion of vertical-strip LLT polynomials. We obtain new expressions for quantities that appear in diagonal harmonics. We consider ∇e n and ∇ωp n , which appear in the Shuffle Theorem and the Square Path Theorem, respectively, and both of which can be expressed using vertical-strip LLT polynomials. As a consequence they are both e-positive after the substitution q → q + 1, see Corollary 6.11 and Theorem 6.14. Finally we discuss chromatic quasisymmetric functions and the representation theory of regular semisimple Hessenberg varieties. In particular Corollary 6.18 provides a new characterisation of chromatic quasisymmetric functions of unit-interval graphs.

Background and main results
We first give additional history and references which lead up to this paper. Then we go into more detail and give all necessary definitions needed in order to state the main results.  in [Hag07,p. 92]) which is a bijection between semi-standard border-strip tableaux and k-tuples of semi-standard Young tableaux of straight shape. They extend their model to include tuples of skew shapes, and they conjecture that LLT polynomials in this family are also Schur positive. Note that the proof of Schur positivity from 2000 does not extend to the Bylund-Haiman model. We remark that a similar model was introduced independently in [SSW03], but they use a more complicated statistic for the q-weight.  [GH06], giving an argument for Schur positivity of LLT polynomials by using Kazhdan-Luztig theory and geometric representation theory. (2007+) J. Haglund [Hag07,Ch. 6] gives an overview of the appearance of LLT polynomials in the study of diagonal harmonics. In particular, the path symmetric functions -which are LLT polynomials indexed by k-tuples of vertical strips -appear in the Shuffle Conjecture [HHL + 05b] and later in the Compositional Shuffle Conjecture [HMZ12], Square Path Conjecture [Ser17] and Delta Conjecture [HRW18] . (2016) In his work on chromatic quasisymmetric functions M. Guay-Paquet [GP16] uses a Hopf algebra approach to show that unicellular LLT polynomials are graded Frobenius series derived from the equivariant cohomology rings of regular semisimple Hessenberg varieties. This yields a second proof that unicellular LLT polynomials are Schur positive, but no combinatorial formula for the coefficients. (2017) E. Carlsson and A. Mellit [CM17] use the so called zeta map on the path symmetric functions to obtain a more convenient model for vertical-strip LLT polynomials. They introduce the Dyck path algebra and prove the Compositional Shuffle Conjecture -henceforth Shuffle Theorem. E. Carlsson and A. Mellit briefly use a plethystic identity originating in [HHL05a]. In the language of Haglund et al. it essentially states that there is a simple plethystic relationship between general fillings and non-attacking fillings. In terms of vertex colorings of graphs this relates a sum over arbitrary colorings to a sum over proper colorings. Consequently there is a plethystic relationship between certain (unicellular) LLT polynomials and the chromatic quasisymmetric functions of J. Shareshian and M. Wachs [SW12,SW16]. 2.2. Symmetric functions. Let Λ denote the algebra of symmetric functions over Q(q). For an introduction to symmetric functions the reader is referred to [Mac96,Sta01]. Elements of Λ may be viewed as formal power series over Q(q) in infinitely many variables x := (x 1 , x 2 , . . . ) that have finite degree and are invariant under permutation of the variables. Alternatively we may think of elements of Λ as Q(q)-linear combinations of the elementary symmetric functions e λ = e λ1 · · · e λr where λ ranges over all integer partitions.
2.3. Schröder paths. Let n := (0, 1) be a north step, e := (1, 0) be an east step, and d := (1, 1) be a diagonal step. A Schröder path P of size n is a lattice path from (0, 0) to (n, n) using steps from {n, e, d} that never goes below the main diagonal, and has no diagonal step on the main diagonal. That is, every east step and every diagonal step of P is preceded by more north steps than east steps. Let S n denote the set of Schröder paths of size n, and let S be the set of all Schröder paths. We describe Schröder paths as words in {n, d, e} * , see Figure 1 for examples. A Schröder path without diagonal steps is called a Dyck path. 2.4. Bounce paths. An important tool in our analysis is a type of bounce path that is similar to bounce paths that appear for example in [GH02,AKOP02]. Given a Schröder path P ∈ S and a point (u 1 , u 0 ) ∈ Z 2 that lies on P define the (partial reverse) bounce path of P at (u 1 , u 0 ) as follows: Starting at (u 1 , u 0 ) move south until you reach the point (u 1 , u 1 ) on the main diagonal. Now move west until you reach a point (u 2 , u 1 ) on P . If this point lies between two diagonal steps of P then continue south until the point (u 2 , u 2 ) on the diagonal. Then move west until you reach a point (u 3 , u 2 ) on P . Continue in this fashion until the bounce path ends at a point incident to a north step or east step of P . An example of a bounce path is shown in Figure 2. The points (u 1 , u 1 ), . . . , (u k , u k ) where the bounce path touches the main diagonal are called bounce points. The coordinates of the peaks (u 1 , u 0 ), (u 2 , u 1 ), . . . , (u k+1 , u k ) of the bounce path are recorded in the bounce partition (u 0 , u 1 , . . . , u k+1 ) of P at (u 1 , u 0 ). There is a unique decomposition where s 1 , s 2 , s 3 , s 4 ∈ {n, d, e} and U, V, W ∈ {n, d, e} * such that U s 1 is a path from (0, 0) to the endpoint (u k+1 , u k ) of the bounce path, and s 4 W is a path from the starting point (u 1 , u 0 ) of the bounce path to (n, n). We call this the bounce decomposition of P at (u 1 , u 0 ). Note that we use a small dot to indicate the starting point and endpoint of the bounce path. In this paper our focus is mainly on the special case where s 1 s 2 ∈ {nn, dn}, V ∈ {n, d} * and s 3 s 4 = de. The pieces that such bounce paths are made of are found in Figure 3. 2.5. The main recursion. An essential step in M. D'Adderio's proof of the epositivity of vertical-strip LLT polynomials is an inductive argument that relies on certain linear relations among them. The starting point of our work is to simplify these relations, and to make them more explicit. This leads to our first main Figure 2. The bounce paths of two Schröder paths P (left) and Q (right) at the point (3, 6). The bounce decompositions are given by P = U n.nV d.eW where U = ∅, V = ddn, and W = ee, and Q = U n.dV d.eW where U = nd, V = n, and W = ee respectively. The left bounce path has two bounce points (3, 3) and (1, 1), and bounce partition (6, 3, 1, 0). result, namely, that the following initial conditions and relations determine a unique symmetric-function valued statistic on Schröder paths.
Theorem 2.1. Let A be a K-algebra that contains Λ. Then there exists a unique function F : S → A, P → F P that satisfies the following four conditions: (i) For all k ∈ N the initial condition F nd k e = e k+1 is satisfied. (ii) The function F is multiplicative, that is, F P Q = F P F Q for all P, Q ∈ S. (iii) For all U, V ∈ {n, d, e} * such that U dV ∈ S we have F U neV − F U enV = (q − 1)F U dV . (iv) Let P ∈ S be a Schröder path, and let (x, z) ∈ Z 2 be a point on P with x+1 < z such that the bounce path of P at (x, z) has only one bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some V ∈ {n, d} * and some st ∈ {nn, dn}. Then Parts of this recursion are not new and have shown up in various forms and shapes in previous research. Due to the plethystic relationship between unicellular LLT polynomials and chromatic quasisymmetric function, some of them have analogs on the chromatic quasisymmetric function side.
Remark 2.2. We call the relation in Theorem 2.1 (iii) the unicellular relation. It can be illustrated by (2) and has appeared for example in [AP18, Ale20, D'A20]. It can be used to express the function F P indexed by a Schröder path as a linear combination of such functions indexed by Dyck paths. Equivalently it can be used to express vertical-strip LLT polynomials as linear combinations of unicellular LLT polynomials. Moreover it is reflected in the definition of the operator ϕ in terms of the operators d + and d − in the Dyck path algebra of E. Carlsson and A. Mellit [CM17].
Remark 2.3. We call relations of the type of Theorem 2.1 (iv) bounce relations. The two bounce relations in (iv) can be illustrated by Theorem 2.1 is formulated in such a way that it only assumes the weakest set bounce relations that are necessary to imply uniqueness. In Section 3 we will see that these relations together with the unicellular relation are strong enough to imply more general bounce relations with arbitrarily many bounce points and bounce paths that may end in different patterns. Moreover Theorem 2.1 remains valid if we allow V ∈ {n, d, e} * .
We prove the uniqueness statement in Theorem 2.1 in Section 3 along with other results of a similar flavour. The existence statement is taken care of in Theorems 2.5 and 2.9. There we show by combinatorial means that there exist two statistics on Schröder paths that satisfy the conditions of Theorem 2.1. We then conclude that they must be equal. It turns out that the unique function F in Theorem 2.1 assigns to each Schröder path the corresponding vertical-strip LLT polynomial.
2.6. Unit-interval graphs. A graph Γ = (V, E) with vertex set V = [n] is a unit-interval graph if uw ∈ E implies uv, vw ∈ E for all u, v, w ∈ [n] with u < v < w. Unit-interval graphs are the incomparability graphs of unit-interval orders. They are in bijection with Dyck paths and of great interest in the context of chromatic (quasi)symmetric functions. The seminal work on this topic is due to J. Shareshian and M. Wachs [SW12,SW16].
A decorated unit-interval graph Γ = (V, E, S) is a pair of a unit-interval graph Γ = (V, E) and a subset S ⊆ E of strict edges. To a Schröder path P in S n we associate a decorated unit-interval graph Γ P with vertex set [n] as follows: For u, v ∈ [n] with u < v there is a non-strict edge uv in Γ P if and only if there is a cell in column u and row v below the path P . Moreover if (u, v) is the endpoint of a diagonal step of P then uv is a strict edge in Γ P . See Figure 4 for an example of this correspondence. The unit-interval graph Γ P . (c) A sligthly simplified diagram (Dyck diagram) describing P . We have that area(P ) = 12, and it is the number of white squares in this diagram. (d) The vertical strips (French notation!) corresponding to P . Thus, ν = (2/1, 3/2, 2/1, 2, 3/1), and G P = G ν , using the Bylund-Haiman convention for indexing LLT polynomials with tuples of skew shapes. The labeling of the boxes indicate the correspondence with the vertices in Γ P . 2.7. Colorings. Let P ∈ S n be a Schröder path. A coloring of Γ P is a map κ : [n] → N + such that κ(u) < κ(v) for all u, v ∈ [n] such that u < v and uv is a strict edge. An ascent of a coloring is a a non-strict edge uv of Γ P such that u < v and κ(u) < κ(v). We let asc(κ) denote the number of ascents of κ.
We remark that our terminology stems from the world of chromatic quasisymmetric functions. Colorings are also closely related to parking functions, and the ascent statistic is called area in that setting, see [Hag07,Ch. 5]. We define the area of P , denoted area(P ) := |E \ S|, as the number of non-strict edges in Γ P .
Example 2.4. Recall the path P = nndnnenedeee from Figure 4. The following diagram illustrates a coloring of P where we have labeled i in Γ P with κ(i): The four edges contributing to asc have been marked with •, and this coloring contributes with q 4 x 2 1 x 2 2 x 3 x 4 x 5 to the sum in (5). 2.8. Vertical-strip LLT polynomials. For P ∈ S n the vertical-strip LLT polynomial G P (x; q) is defined as where the sum ranges over all colorings of Γ P . One can show that G P (x; q) is a symmetric function and a straightforward proof of this fact is given in [AP18], which is an adaptation of the proof given in the appendix of [HHL05a]. Alternatively this also follows from the e-expansion provided in this paper. Every LLT polynomial indexed by a k-tuple of vertical-strip skew shapes in the Bylund-Haiman model can be realized as G P (x; q) for some P ∈ S n , see [CM17,AP18]. The family of vertical-strip LLT polynomials indexed by Dyck paths are referred to as unicellular LLT polynomials.
The vertical-strip LLT polynomials satisfy the recursion in Theorem 2.1.
Theorem 2.5. Let F : S → Λ be defined by F P = G P . Then F satisfies the conditions in Theorem 2.1 (i)-(iii). Moreover let P ∈ S be a Schröder path, and let (x, z) ∈ Z 2 be a point on P with x + 1 < z such that the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some st ∈ {nn, dn, nd}. Then A similar result was shown by M. D'Adderio in [D'A20] in the language of the Dyck path algebra. D'Adderio in turn built on the results that E. Carlsson and A. Mellit derived in their proof of the Shuffle Theorem [CM17]. In Section 4 we give an alternative proof of the fact that the vertical-strip LLT polynomials satisfy the conditions of Theorem 2.1. This proof makes use of the combinatorics of colorings of unit-interval graphs and bijective arguments rather than relations between operators in the Dyck path algebra. Note that Theorem 2.5 contains more general relations than Theorem 2.1. In particular it contains a third type of bounce relation which is illustrated by U n.dV d.eW A proof of Theorem 2.5 is achieved by combining the results of Section 4 with Proposition 3.1, which deduces the third bounce relation from the first two, and with Theorem 3.3, which yields bounce relations with arbitrarily many bounce points. Theorem 2.5 clearly implies the existence statement in Theorem 2.1. Moreover Theorems 2.1 and 2.5 provide a new characterization of vertical-strip LLT polynomials.
Corollary 2.6. The vertical-strip LLT-polynomials G P are uniquely determined by the initial condition and relations in Theorem 2.1.
In this paper we make use of this characterization to prove an explicit positive expansion of G P (x, q + 1) in terms of elementary symmetric functions.
2.9. Orientations. Let Γ = (V, E) be a graph. An orientation of Γ is a function θ : E → V 2 that assigns to each edge uv ∈ E a directed edge → uv or → vu. Alternatively, we may view θ = {θ(e) : e ∈ E} ⊆ V 2 as a set that contains a unique directed edge for each edge in E. We adopt the convention of viewing orientations as sets, which is more convenient if we want to add or remove edges. Let Γ = (V, E ) be a subgraph of Γ, that is, E ⊆ E. An orientation θ of Γ yields an orientation θ of Γ defined by We call θ the restriction of θ to E . If V = [n] then Γ is equipped with the natural orientation that assigns to each edge uv the directed edge → uv where u < v. Let P ∈ S n and let Γ P = (V, E, S) be the corresponding decorated unit-interval graph. We let O(P ) denote the set of orientations θ of Γ P such that the restriction of θ to S is the natural orientation. Given θ ∈ O(P ), an edge → uv ∈ θ is an ascending edge in θ if u < v and uv / ∈ S. We let asc(θ) denote the number of ascending edges in θ.
Example 2.7. We use the diagram notation as in Figure 4, to illustrate an orientation θ. The ascending edges in θ are marked with → in the diagram, and asc(θ) = 5. 2.10. The highest reachable vertex. Let θ ∈ O(P ). For a vertex u of Γ P the highest reachable vertex , denoted hrv(θ, u), is defined as the maximal v such that there is a directed path from u to v in θ using only strict and ascending edges. By definition hrv(θ, u) ≥ u for all u. The orientation θ defines a set partition π(θ) of the vertices of Γ P , where two vertices belong to the same block if and only if they have the same highest reachable vertex. Finally, let λ(θ) denote the integer partition given by the sizes of the blocks in π(θ).
2.11. An explicit e-expansion. Given a Schröder path P ∈ S n define the symmetric functionĜ P (x; q) bŷ The second main result of this paper states that the symmetric functions defined in (7) satisfy the initial conditions and relations in Theorem 2.1.
Theorem 2.9. Let F : S → Λ be defined by F P =Ĝ P . Then F satisfies the conditions in Theorem 2.1.
Combining Theorems 2.1, 2.5 and 2.9 we obtain the first explicit expansion of the vertical-strip LLT polynomials in terms of elementary symmetric functions.
Corollary 2.10. For all P ∈ S we have G P =Ĝ P . In particular G P (x; q + 1) expands positively into elementary symmetric functions.
The positivity result in Corollary 2.10 was obtained by M. D'Adderio in [D'A20]. The formula in (7) was conjectured in [GHQR19] and independently by the first named author in [Ale20]. Combinatorial positive expansions into elementary symmetric functions were obtained in a few special cases in [AP18,Ale20], albeit with a different statistic in place of λ(θ).
It is worth noting that the right-hand side of (7) is manifestly symmetric. In contrast, previously known formulas for LLT polynomials in terms of power-sum symmetric functions or Schur functions (with signs) rely on the expansion into fundamental quasisymmetric functions. To derive these formulas one needs to use the fact that LLT polynomials are symmetric in the first place, and not just quasisymmetric.
Remark 2.11. A stronger version of (7) has been proved recently in [AN20b,AN20a]. The authors mainly consider the unicellular case, and they use the modular law and its dual in their proof.

Bounce relations
In this section we prove the uniqueness statement in Theorem 2.1. In a first step we show that the bounce relations in Theorem 2.1 (iv) are equivalent to, and hence imply, other types of bounce relations. This is the content of Proposition 3.1. We then prove the presence of bounce relations with arbitrarily many bounce points in Theorem 3.3. With these more general relations at our disposal the proof of the uniqueness statement can be carried out. We conclude the section with a characterization of unicellular LLT polynomials that uses bounce relations for Dyck paths in Theorem 3.5.
Recall that we use dots in the bounce decomposition to highlight the starting point and endpoint of the bounce path in consideration. Note that the location of these dots does not change the word encoding a path. Moreover, slightly different choices of starting points may result (essentially) in the same bounce path. For example, U ne.nV en.eW and U ne.nV e.neW describe the same Schröder path, and the indicated bounce paths have all bounce points in common.
Proposition 3.1. Let A be a Q(q)-algebra, and let F : S → A, P → F P be a function that satisfies the unicellular relation, that is, Then the following additional sets of relations are equivalent: (iv) Schröder Relations A. Let P ∈ S be a Schröder path, and let (x, z) ∈ Z 2 be a point on P with x + 1 < z such that the bounce path of P at (x, z) has only one bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some st ∈ {nn, dn}. Then (v) Schröder Relations B. Let P ∈ S be a Schröder path, and let (x, z) ∈ Z 2 be a point on P with x + 1 < z such that the bounce path of P at (x, z) has only one bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some st ∈ {nn, nd}. Then (vi) Dyck Relations. Let P ∈ S be a Schröder path, and let (x, z) ∈ Z 2 be a point on P with x+1 < z such that the bounce path of P at (x, z) has only one bounce point. If the bounce decomposition of P at (x, z) is given by P = U n.nV ne.eW then If the bounce decomposition of P at (x, z) is given by P = U ne.nV ne.eW then Remark 3.2. Curiously (13) is reminiscent of a cross-product and can be expressed as the following vanishing determinant, where multiplication is now replaced by concatenation: Expanding this determinant according to Sarrus' rule gives exactly the terms in (13) with correct sign.
Proof of Proposition 3.1. We first show that (8) and (10) are equivalent to (12). We start with (12), Bringing one term to the other side we obtain F U n.nV ne.eW − F U n.nV en.eW = q(F U n.nV en.eW − F U n.nV e.enW ).
Applying the unicellular relation (iii) on both sides we obtain which after the cancellation of a common factor becomes (8). Clearly each step is reversible and we obtain the desired equivalence.
Next we show that (9) is equivalent to (13). We start with (13), which is rearranged as U ne.nV ne.eW Applying the unicellular relation twice on the right-hand side we get After additional applications of the unicellular relation on the left-hand side we get (after dividing by q − 1) Rearranging the terms again we arrive at which after a final application of the unicellular relation on both sides gives (9).
To complete the proof we now show that (11) is equivalent to (13) assuming that (10) and (12) are valid. We start with (11) which becomes U n.dV d.eW after moving one term to the left-hand side. We now use the unicellular relation to eliminate one diagonal step in each path. In the left-hand side, we keep the first diagonal step and eliminate the second one. In the right-hand side we keep the second diagonal step. After cancelling a common factor of (q − 1), the left-hand side simplifies to Similarly the right-hand side becomes Now we use (10) on both terms in the right-hand side, that is, qF U n.neV e.dW = F U n.neV d.eW and qF U en.nV e.dW = F U en.nV d.eW . Moving all terms to the left-hand side, we obtain the relation Finally, we eliminate the remaining diagonal steps using the unicellular relation. After cancellation of a common factor this gives the identity 0 = (F U en.nV ne.eW − F U en.nV e.neW ) − F U n.neV ne.eW − F U n.neV e.neW + F U n.neV ne.eW − F U ne.nV ne.eW − (F U n.neV e.enW − F U ne.nV e.enW ) .
After cancelling the underlined terms we arrive at (13). Since all steps in this deduction are invertible we have the desired equivalence.
Our next goal is to show that the relations in Theorem 2.1 are strong enough to impose relations involving bounce paths with arbitrarily many bounce points.
Theorem 3.3. Let A be a Q(q)-algebra, and let F : S → A, P → F P be a function that satisfies Conditions (iii) and (iv) in Theorem 2.1. Let P ∈ S be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some V ∈ {n, d} * and some st ∈ {nn, nd, dn}. Then Proof. The claim is shown by induction on the number of bounce points of the bounce path of P at (x, z). First assume that the bounce path of P at (x, z) has only one bounce point. If st ∈ {nn, dn} then there is nothing to show. If st = nd then the claim follows from Proposition 3.1. To see this note that the proof of the equivalence of Proposition 3.1 (iv) and (v) goes through in the same way if we impose the restriction V ∈ {n, d} * in both cases. Thus the base case of the induction is taken care of. Otherwise the bounce partition of P at (x, z) has length r > 3 and is given by (z, x, v, . . . ). Set y = x + 1 and w = v + 1. In this case there are at least two bounce points and we may write . Note that here we use the fact that z > y, which also implies x > w. By the unicellular relation we have (16) To simplify notation suppose for the moment that st = dn. By applying first Theorem 2.1 (iv) at the point (x, z), and then the induction hypothesis at the point (v, y) we obtain Note that the bounce path of U s.tV ndeV e.dW at (v, y) has the same bounce points as the bounce path of P at (x, z) except for (x, x). Thus the induction hypothesis can be applied. Similarly, by applying first induction hypothesis at the point (v, x), and then Theorem 2.1 (iv) at the point (x, z) the we obtain (20) Note that the bounce path of U s.tV denV e.dW at (v, x) has the same bounce points as the bounce path of P at (x, z) except for (x, x). Thus the induction hypothesis can be applied.
Combining (15), (17) and (19), and using the unicellular relation one more time, we obtain This proves the claim in the case st = dn. The other two cases st ∈ {nn, nd} follow in the exact same manner.
The uniqueness in Theorem 2.1 can now be shown using an intricate induction argument devised by M. D'Adderio in the proof of [D'A20, Thm. 5.4]. Given a Schröder path P our strategy is to show that F P can be expressed in terms of elementary symmetric functions as well as functions F Q where Q has size less than P , or Q has fewer east steps than P , or Q is obtained from P by moving an east step of P to the left.
Proof of uniqueness in Theorem 2.1. Let P be a Schröder path of size n with r east steps, and assume that P = XeW where X ∈ {n, d} * defines a path from (0, 0) to (x, z), and W ∈ {n, d, e}. That is, X is the initial segment of P that leads up to the first east step of P .
We first use induction on the size n. By Theorem 2.1 (i) the only path of size 1, namely ne, is sent to 1 by F . Thus we may assume that n > 1 and that F Q is uniquely determined for all Q ∈ S m for all m < n.
Secondly we use induction on r. If P has only one east step, then P is of the form P = nd n−1 e and F P = e n (x) is determined by the initial condition in Theorem 2.1 (i). Thus the base case of the second induction is taken care of. We may assume that P has at least two east steps, and that F Q is determined uniquely for all Schröder paths Q ∈ S n with fewer than r east steps.
Thirdly we use induction on z. The base case for the third induction is z = x + 1. If z = x + 1 then Xe and W are Schröder paths of size less than n. Note that here we use the fact that P contains two or more east steps. By Theorem 2.1 (ii) and by induction on n it follows that F P = F Xe F W is determined uniquely. Thus we may assume that z > x + 1 and that F Q is determined uniquely for all Q ∈ S n with r east steps that are of the form Q = X eW where X ∈ {n, d} * defines a path from (0, 0) to (x , z ) with z < z. Let L ⊆ S n denote the set of all such Schröder paths.
Now if X = Y n, that is P = Y neW , then Y dW is a Schröder path of size n with fewer east steps than P . Note that here we use the fact that z > x + 1. Similarly Y enW is a Schröder path in L. By Theorem 2.1 (iii) and by induction on r and z it follows that If on the other hand X = Y d, that is P = Y deW , then let P = U s.tV d.eW be the bounce decomposition of P at (x, z). Clearly V ∈ {n, d} * and st ∈ {nn, nd, dn}. Thus by Theorem 3.3 we may write F P as a linear combination of functions F Q for certain paths Q ∈ L. Therefore F P is determined uniquely by induction on z. This completes the proof.
Remark 3.4. The conditions in Theorem 2.1 were chosen because they are both natural and relatively easy to verify in the following sections. However, there are various equivalent formulations that might be more practical in other contexts. By Proposition 3.1 one can replace the relations in Theorem 2.1 (iv) by one of the other sets of bounce relations given in Proposition 3.1 (v) or (vi). Moreover, the multiplicativity condition Theorem 2.1 (ii) can be dropped if one replaces Theorem 2.1 (i) by the stronger initial condition that F P = e λ for all partitions λ, where P = nd λ1−1 end λ2−1 e · · · nd λ −1 e. In this case a similar induction argument as in the proof above works. We refer again to the proof of [D'A20, Thm. 5.4].
Moreover we can prove a Dyck path analog of Theorem 2.1. Let D denote the set of all Dyck paths, that is, the set of Schröder paths that contain no diagonal steps. Note that using our conventions, if P is a Dyck path and (x, z) is a point on P then the bounce path of P at (x, z) must have a single bounce point.
Theorem 3.5. The function that assigns to each Dyck path the corresponding unicellular LLT polynomial is the unique function F : D → Λ, P → F P that satisfies the following conditions: (i) For all k ∈ N there holds the initial condition (vi) Let P ∈ D be a Dyck path and let (x, z) ∈ Z 2 be a point on P with x + 1 < z.
If the bounce decomposition of P at (x, z) is given by P = U n.nV ne.eW then F U n.nV ne.eW = (q + 1)F U n.nV e.neW − qF U n.nV e.enW .
If the bounce decomposition of P at (x, z) is given by P = U ne.nV ne.eW then F U en.nV en.eW + F U ne.nV ne.eW + F U n.neV e.enW = F U en.nV ne.eW + F U ne.nV e.enW + F U n.neV e.neW .
Proof. Suppose that F : D → Λ satisfies the conditions in the theorem. First note that F can be extended to a functionF : S → Λ on Schröder paths in a unique way such that the unicellular relation Proposition 3.1 (iii) holds. MoreoverF is multiplicative on Schröder paths. We next show that a functionF satisfies the relations Proposition 3.1 (vi). To see this let P be a Schröder path and (x, z) ∈ Z 2 be a point on P with x + 1 < z, such that the bounce path of P at (x, z) has only one bounce point, and such that the bounce decomposition of P at (x, z) is of the form P = U n.nV ne.eW for some U, V, W ∈ {n, d, e} * . Using the unicellular relation (many times) we can writeF P as a linear combination of symmetric functions ofF Q = F Q , where Q is a Dyck path of the form Q = U n.nV ne.eW . In particular the bounce paths of P and Q at (x, z) coincide. We may now apply the linear relation from the assumption to the symmetric functions F Q and use the unicellular relation to reintroduce the diagonal steps. In this way we have show thatF satisfies (12) and the same can be done for (13).
It now follows from Proposition 3.1 and Theorem 2.1 that the functionF , and in particular the function F must be unique. Moreover, by Theorem 2.5 and Proposition 3.1 the function that assigns to each Dyck path the corresponding unicellular LLT polynomial satisfies Conditions (ii) and (vi) above. The fact that this function also satisfy the initial condition in (i) is an easy consequence of the e-expansion in Corollary 2.10 1 . This completes the proof.

Bijections on colorings
In this section we prove Theorem 2.5. We first verify Theorem 2.1 (i)-(iii) in Proposition 4.2. We then proceed to demonstrate the two bounce relations of Theorem 2.1 (iv) in Theorems 4.4 and 4.5. The method of proof is purely bijective and similar to that in [Ale20, Prop. 18]. Indeed, (iii) as well as the st = nn case of (iv) were already proved therein. To obtain the full claim of Theorem 2.5 one can then use the results of Section 3, namely Proposition 3.1 and Theorem 3.3. We remark that it is also possible to give a bijective proof for the third type of bounce relation in Theorem 2.5, that is, the case st = nd, if there is only one bounce point. However, Theorem 3.3 is currently our only way to prove bounce relations with many bounce points.
In the proof of Theorem 3.3, we used diagrams to get a good overview of the recursions. In this section we use such diagrams to denote weighted sums over vertex colorings of decorated unit-interval graphs. To simplify the notation, we shall only show the vertices and edges of the diagrams which matter. The following example introduces the necessary conventions.
Example 4.1. The LLT polynomials G P (x; q) and G Q (x; q) for P = nndneee and Q = nnddee are given as sums over vertex colorings. We let A(x; q) be the sum over colorings of Γ P , with the extra condition that κ(2) ≥ κ(4). Expressed using diagrams, we have Note that we use ⇓ to impose a weak inequality. That is, if an edge xy is marked with ⇓, then we require that κ(x) ≥ κ(y). In the same manner, ↓ is used to indicate a strict inequality which is a non-ascent. We utilize the convention that edges in gray boxes do not contribute to the ascent statistic (this is consistent with marking all strict edges gray). Moreover, edges of particular importance are marked with a center-dot; we shall possibly refine our argument depending on if the colorings under consideration have this edge as ascending or not. Now, a simple bijective argument shows that since a coloring on the left-hand side appears as a coloring of exactly one of the diagrams in the righ hand side. Note that it is only the edge from 2 to 4 that really matters. Stated in full generality, for any edge → wy in any diagram, we have the identity This simply expresses the fact that a set of colorings κ can be partitioned into two sets, the first set where κ(w) > κ(y) and the second set where κ(w) ≤ κ(y).
1 The e-expansion of the unicellular LLT polynomial of the path graph was also obtained in [AP18,Prop. 5.18] Note that most diagrams we use do not represent vertical-strip LLT polynomials, but merely a sum κ q asc(κ) x κ(1) · · · x κ(n) , where the sum ranges over colorings compatible with the restrictions imposed by the diagram.
Proposition 4.2. Let F : S → Λ be the function that assigns to each Schröder path P the symmetric function F P = G P (x; q) defined in (5).
(i) Then F satisfies the initial condition F nd k e = e k+1 for all k ∈ N.
(ii) The function F is multiplicative, that is, Proof. For (i), we have by definition that F nd k e is a sum over colorings κ : [k + 1] → N + , such that κ(1) < κ(2) < · · · < κ(k + 1). Now it is clear that we indeed obtain e k+1 . Moreover, (ii) is immediate from the definition, as the graph Γ P Q is (up to a relabeling of the vertices) the disjoint union of the graphs Γ P and Γ Q . Colorings can thus be done on each component independently and the statement follows.
It remains to prove (iii). Using diagrams as in (21), this identity can be expressed as follows: By splitting the first and the last diagram into subcases depending on whether κ(w) < κ(y) or κ(w) ≥ κ(y) as in (21), it becomes evident that the identity holds.
In the remainder of this section we assume that x + 1 = y so that they are adjacent entries on the diagram diagonal. 4.1. Bounce relations for colorings. We need a simple bijection between certain colorings which is referred to as the swap map. The swap map sends a coloring κ to the coloring κ defined as This map is used in all further proofs in this section. Lemma 4.3 (Swap Map). Let P ∈ S and let (x, z − 1) ∈ Z 2 be a point on P such that the bounce path of P at (x, z − 1) has one bounce point, and such that the bounce decomposition of P at (x, z − 1) is given by P = U n.nV e.eW . Set y = x + 1. Then where we sum over all colorings of P on both sides, with the indicated added restrictions on the colors of x and y.
Proof. Expressing the statement as an identity of diagrams, we have In order to prove (23) we show that the swap map κ → κ preserves the number of ascents, not counting the contribution of the edge xy. Note that for all j with w < j < x or y < j < z we have that This implies that for any coloring κ appearing in the left-hand side of (23), we have that asc(κ) = 1 + asc(κ ).
We proceed by showing that the vertical-strip LLT polynomials satisfy the two bounce relations in Theorem 2.1.
Theorem 4.4 (The first bounce relation for colorings). Let P ∈ S be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce path of P at (x, z) has a single bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U n.nV d.eW . Then G P (x; q) = qG U n.nV e.dW (x; q).
Proof. We want to prove the following identity expressed as diagrams: If → xz and → yz are ascending edges on both sides, we use the identity map on the set of colorings. By using the transitivity of the forced inequalities, it then suffices to prove the identity But this follows from Lemma 4.3.
Theorem 4.5 (The second bounce relation for colorings). Let P ∈ S be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce path of P at (x, z) has a single bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U d.nV d.eW . Then G P (x; q) = G U n.dV e.dW (x; q).
Proof. We want to prove the identity expressed as diagrams in (4). For convenience, it is presented here as follows: We split both sides into cases according to (21): The first term on both sides cancel, and there are no colorings compatible with the inequalities in the last term on both sides. Moreover, transitivity forces the last inequality in the remaining cases: All four remaining diagrams have the same q-weight, namely one fixed ascent (recall the convention for shaded boxes). The swap map sends the terms on the left-hand side to the terms on the right-hand side. This concludes the proof.

Bijections on orientations
In this section we prove Theorem 2.9. To be precise we first treat the easier claims in Proposition 5.2. We then show that the symmetric functionsĜ P (x; q) satisfy the bounce relations of Theorem 2.1 (iv) in Theorems 5.4 and 5.17. Since we want to work with orientations we actually show that the symmetric functionsĜ P (x; q + 1) satisfy a shifted version of these relations. Our proofs are purely bijective and the involved bijections are very simple. However, the proofs of the bounce relations are a bit tedious in the sense that they require the distinction of quite a few different cases. We remark that the third type of bounce relations in Theorem 2.5, ending in a pattern st = nd, can be given a bijective proof in similar style, assuming that there is only one bounce point.
As with the colorings, we use diagrams to illustrate the different cases. However, in this section each diagram represents a sum over orientations θ weighted by q asc(θ) e λ(θ) .
Example 5.1. The symmetric functionĜ Q (x; q +1) where Q = nnddee is described by the diagramĜ Q (x; q + 1) = → 4 → 3 2 1 which represents a sum over eight orientations (all possible ways to orient the edges 12, 23 and 34). By specifying additional edges in the diagram, we obtain a sum over a smaller set of orientations. For example, the two following diagrams represent the indicated sums.
Notice that in the second diagram, we again use the convention that edges marked gray do not contribute to ascents.
We start out by proving that the function P →Ĝ P (x; q + 1) satisfies the correct initial conditions, is multiplicative, and obeys the shifted version of the unicellular relation.
Proposition 5.2. Let F : S → Λ be the function that assigns to each Schröder path P the symmetric function F P =Ĝ P (x; q + 1) defined in (7).
(i) Then F satisfies the initial condition F nd k e = e k+1 for all k ∈ N.
(ii) The function F is multiplicative, that is, Proof. To see (i) let P = nd k e ∈ S k+1 . Note that the decorated unit-interval graph Γ P has k strict edges and no other edges. Thus O(P ) contains a unique orientation θ which has no ascents. Its highest reachable vertex partition is given by λ(θ) = (k + 1) since the vertex k + 1 can be reached from every other vertex using the strict edges. Hence F P = e k+1 as claimed.
To see (ii) let P, Q ∈ S. Note that Γ P Q is (up to a relabeling of the vertices) the disjoint union of the graphs Γ P and Γ Q . Thus every orientation of Γ P Q is (again up to a relabeling) the disjoint union of an orientation of Γ P and Γ Q . It follows that To see (iii) let U, V ∈ {n, d, e} * such that U dV ∈ S. Let wy, where w < y, denote the strict edge in Γ U dV corresponding to the diagonal step between U and V . Given that preserves highest reachable vertices. Moreover asc(φ(θ)) = asc(θ) − 1 if → wy ∈ θ, and asc(φ(θ)) = asc(θ) if → yw ∈ θ. The claim follows.
We now turn our attention to the bounce relations. To begin with we need some additional definitions. Given a finite totally ordered set X = {u 1 < u 2 < · · · < u r } and a function σ : X → N + , we identify σ with the word σ 1 σ 2 . . . σ r where σ i = σ(u i ) for all i ∈ [r]. Note that this word contains all the information on σ if the domain X is known. The weak standardisation of σ is defined as the unique surjection wst(σ) : X → [m] that satisfies for all u, v ∈ X, where m is (necessarily) the cardinality of the image of σ.
Let O(n) denote the set of orientations of unit-interval graphs on n vertices. Given a Schröder path P ∈ S n and a set of directed edges D ⊆ Throughout the remainder of this section let P be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce partition of P at (x, z) is given by (z, x, v), and the bounce decomposition of P at (x, z) is given by P = U s.tV d.eW for some st ∈ {nn, dn} and U , V , W ∈ {n, d, e}. In particular, the bounce path of P at (x, z) has only one bounce point (x, x). Set w := v + 1 and y := x + 1, and Let τ := (x, y) ∈ S n be a transposition. Given A ⊆ O(n) and a function f : A → O(n), we say that f is π-preserving on A if π(f (θ)) = π(θ) for all θ ∈ A. The function f is called π-switching on A if π(f (θ)) = τ (π(θ)). Both of these properties imply that f is λ-preserving on A, that is λ(f (θ)) = λ(θ) for all θ ∈ A. To see this note that permuting the entries of a set partition does not change the block structure. Define a function by letting ρ y (θ) be the maximal vertex in [n] that can be reached from y using only ascending and strict edges in θ without using the edge → yz. Note that ρ w (θ) ≤ hrv(θ, w), ρ x (θ) ≤ hrv(θ, x) and ρ y (θ) ≤ hrv(θ, y).
5.1. The first bounce relation for orientations. We now take care of the case st = nn. Our goal is to prove the following theorem.
Theorem 5.4. Let P ∈ S be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce path of P at (x, z) has a single bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U n.nV d.eW . Set Q := U n.nV e.dW . Then G P (x; q + 1) = (q + 1)Ĝ Q (x; q + 1).
A proof of Theorem 5.4 is given at the end of this subsection. Our strategy is to find a λ-preserving two-to-one function Φ : O(P ) → O(Q) such that the fibres of Φ satisfy θ∈Φ −1 (θ ) q asc(θ) = (q + 1)q asc(θ ) for all θ ∈ O(Q). We build such a function Φ from three building blocks φ, φ , and ψ defined below. The following result is immediate from this definition.
If we impose restrictions via σ we can say more. is a π-preserving bijection.
We now collect the orientations on which we plan to use the map φ in the proof of Theorem 5.4. q asc(θ) = (q + 1)q asc(θ ) .
for all θ ∈ B.
Consequently φ : A → B is well-defined and π-preserving on A.
We now collect the orientations on which we plan to use the map φ in the proof of Theorem 5.4. Proof. In each case σ satisfies σ 1 ≥ σ 2 = σ 3 . Hence Lemma 5.11 applies. Each codomain is in bijection with two distinct domains depending on a choice of a. The claim on the q-weights follows from Lemma 5.10.

Next let ψ : O(P ) → O(Q) be the function defined by
for all i ∈ {w, . . . , x − 1} ∪ {y + 1, . . . , z − 1}, and for all other edges. In words, ψ(θ) is obtained from φ(θ) by exchanging the rows x and y, and the columns x and y (excluding the edge xy and the strict edge yz), and by forcing the inclusion of  If we impose restrictions on the data we can say more.
Proof. Let A = O(P, σ; a, b) and B = O(Q, σ 2 σ 1 σ 3 ; → yx). We first show that ψ : A → O(Q) is π-switching. This implies that ψ : A → B is well-defined, that is, that ψ really maps A to B. The injectivity of ψ follows from Lemma 5.14. To conclude the proof we then show that ψ : A → B is also surjective.
We now collect the orientations on which we plan to use the map ψ in the proof of Theorem 5.4. q asc(θ) = (q + 1)q asc(θ ) for all θ ∈ B.
Proof. The data σ, a, b in each of the domains satisfies Conditions (i) and (ii) in Lemma 5.15. Hence the lemma applies. Each codomain is in bijection with two distinct domains depending on a choice of a or b. The claim on the q-weights follows from Lemma 5.14.
The results of this section combine to a proof of the desired bounce relation among the symmetric functionsĜ P (x; q + 1).
Proof of Theorem 5.4. Define a function Φ : O(P ) → O(Q) by letting according to Table 2. The reader should verify that the table covers all orientations in O(P ) and O(Q). Lemmas 5.8, 5.12 and 5.16 show that Φ is a λ-preserving two-to-one function such that the fibres of Φ satisfy θ∈φ −1 (θ ) q asc(θ) = (q + 1)q asc(θ ) for all θ ∈ O(Q). The claim follows. 5.2. The second bounce relation for orientations. We now treat the case st = dn. Our goal is to prove the following theorem.
Theorem 5.17. Let P ∈ S be a Schröder path and let (x, z) ∈ Z 2 be a point on P with z > x + 1 such that the bounce path of P at (x, z) has a single bounce point, and such that the bounce decomposition of P at (x, z) is given by P = U d.nV d.eW . Set Q := U n.dV e.dW . Then G P (x; q + 1) =Ĝ Q (x; q + 1).
A proof of Theorem 5.17 is given at the end of this subsection. Our strategy is to find a λ-preserving bijection Φ : O(P ) → O(Q) such that the asc(Φ(θ)) = asc(θ) for all θ ∈ O(P ). The bijection Φ is made up of two building blocks φ and ψ.
Let φ : O(P ) → O(Q) be the function defined by In words, φ(θ) is obtained from θ by adding and removing some strict edges as needed, by demanding → xv ∈ φ(θ) if and only if → yz ∈ θ, and by leaving all other edges unchanged.
It follows that φ : A → B is well-defined and π-preserving on A.
We now collect the orientations on which we plan to use the map φ in the proof of Theorem 5.17. Proof. All cases satisfy the conditions of Lemma 5.20. The claim on the ascents follows from Lemma 5.19.
Next we define a second map that covers the remaining cases. Let ψ : for all other edges. In words, ψ(θ) is obtained from θ by exchanging the rows x and y, and the columns x and y, except that the edges vx, xy and yz (as well as some strict edges) have to be treated separately.
Example 5.22. As seen in the figure here, the map ψ swaps the marked columns above x and y, and the marked rows to the left of x and y. It is a bijection for all choices of α, β ∈ {→, ↓}.
It is immediate from the definition that this map is invertible and preserves the number of ascents.
If we impose restrictions on σ we can say more.
Proof. Let A = O(P, σ; b, c) and B = O(Q, σ 2 σ 1 σ 3 ; a, b ). We first show that ψ : A → O(Q) is π-switching. By Lemma 5.23 this implies that ψ : A → B is injective and well-defined, that is, that ψ really maps A to B. To conclude the proof we then show that ψ : A → B is also surjective.
This completes the proof.
We now collect the orientations on which we plan to use the map ψ in the proof of Theorem 5.17.
according to Table 3. The reader should verify that the table covers all orientations in O(P ) and O(Q). Lemmas 5.21 and 5.25 provide that Φ is a λ-preserving bijection that preserves the number of ascents. The claim follows.

Applications and further directions
In this final section we briefly visit some areas that are related to our results, and collect several consequences and open problems. We hope that readers familiar with some but not all of these topics can use it as a starting point for learning more.
6.1. Schur expansions. It is a major open problem to find an explicit positive formula for the Schur expansion of G P (x; q). In this subsection we compare a new signed Schur expansion with the previously known signed Schur expansion. The following description is a straightforward application of techniques in [HHL05a], and appears in [GHQR19, Eq. 3.16]. Given a permutation σ ∈ S n , let the inverse ascent set ASC(σ) be defined as Let P be a Schröder path of size n and let S n (P ) denote the set of colorings σ of P such that (σ(1), σ(2), . . . , σ(n)) is a permutation in S n . The Egge-Loehr-Warrington theorem [ELW10] yields the Schur expansion Note that α(σ) is a composition in general. The expression s α(σ) should be computed via the Jacobi-Trudi identity and simplifies either to 0, or a Schur function with a sign.
The results in this paper yield a new Schur expansion of vertical-strip LLT polynomials.
Corollary 6.2. The Schur expansion of G P (x; q) is given by where K µλ is a Kostka coefficient.
One advantage of (36) over the formula in (35) is that is provides a more explicit formula for the coefficient of a specific Schur function. Moreover, it might be easier to come up with a sign-reversing involution on the objects in (36) which solves the Schur-positivity problem. Finally we note that the Schur coefficients of hook shapes in the unicellular LLT polynomials are determined in [HNY20]. Proof. Since the LLT polynomials G P and G rev(P ) are homogeneous symmetric functions of degree n, it is enough to show that G P (x 1 , x 2 , . . . , x n ; q) = G rev(P ) (x n , . . . , x 2 , x 1 ; q).
As a corollary, vertical-strip LLT polynomials satisfy the dual bounce relations. These are the relations obtained from the bounce relations in Theorem 2.5 by reversing all paths.
Corollary 6.5. The function F : S → Λ defined by P → G P satisfies the relations obtained from the bounce relations in Theorem 2.5 by reversing all paths. Moreover this function is uniquely determined by the conditions in Theorem 2.1 (i)-(iii) together with the dual bounce relations obtained by reversing all paths in Theorem 2.1 (iv) (or, alternatively, in Proposition 3.1 (v)).
Corollary 6.6. The function F : D → Λ defined by P → G P is uniquely determined by the conditions in Theorem 3.5 (i) and (ii) together with the dual bounce relations obtained by reversing all paths in Theorem 3.5 (vi).
Note that some sets of relations we have encountered in this paper are self-dual , that is, invariant under reversal of all underlying paths. Examples are the unicellular relation and the second type of bounce relations in Theorem 2.5 -the case st = dn. Moreover the initial conditions in Theorem 2.1 (i) and Theorem 3.5 (i) are self-dual. However Lemma 6.4 is not manifestly evident, neither from the characterization of vertical-strip LLT poylnomials in Theorem 2.1, nor from the e-expansion in Corollary 2.10.
Open Problem 6.7. Give a combinatorial explanation of the identity Open Problem 6.8. Deduce the dual bounce relations directly from the bounce relations (in the style of the proof of Proposition 3.1).
Moreover it is an interesting question if there are other sets of relations that uniquely determine the vertical-strip LLT polynomials.
Open Problem 6.9. Combining bounce relations and dual bounce relations, can we find other results in the style of Theorem 2.1?
An answer to this question can be found in the recent [AN20a], where the Dyck relation and the dual is shown to uniquely define the chromatic quasisymmetric functions.
6.3. Hall-Littlewood polynomials. For a Dyck path P it is noted in [AP18] that G P (x; q) = G rev(P ) (x; q) and that ωG P (x; q) = q area(P ) G P (x; q −1 ), and that a similar relation holds for general Schröder paths, where vertical-strip LLT polynomials are mapped to horizontal-strip LLT polynomials. Hence, there is a formula analogous to (7) for the horizontal-strip LLT polynomials. As a special case, we can say something about the transformed Hall-Littlewood polynomials.
The transformed Hall-Littlewood polynomial H µ (x; q) may be defined as follows (see [TZ03,Eq. (2)+(11)] for a reference). For λ n let where the R ij are raising operators acting on the partitions (or compositions) indexing the complete homogeneous symmetric functions by Hence, our main result implies that It would be interesting to see if there is a direct combinatorial proof of this identity connecting Kostka-Foulkes polynomials and ascents. Moreover, perhaps it is possible to interpret the raising operators used to define the transformed Hall-Littlewood polynomials on the level of orientations.
6.4. Diagonal harmonics. We shall briefly sketch some consequences in diagonal harmonics. For more background on this topic, we refer to [Ber09,Hag07]. Let us first recall the definition of the operators ∇ and ∆ f . They act on the space Λ(q, t) of symmetric functions over Q(q, t), and the modified Macdonald polynomials are eigenvectors. First set B(µ) := (i,j)∈µ (q i−1 t j−1 ), where the sum ranges over all cells in the Young diagram of µ. We then define the linear operators ∇ and ∆ f by where we use plethystic substitution. For example, e 2 [B(2, 1)] = e 2 [1 + q + t] = e 2 (1, q, t) = q + t + qt.
Let F n (x; q, t) := ∇e n . M. Haiman showed [Hai94] that F n (x; q, t) is the bigraded Frobenius series of the space of diagonal coinvariants in two sets of n variables where S n act diagonally. A great deal of research has been devoted to give combinatorial interpretations of ∇f , for various choices of symmetric function f .
In [HHL + 05b], the authors presented a conjectured combinatorial formula for F n (x; q, t), the Shuffle Conjecture. This conjecture was later refined in [HMZ12], where the Compositional Shuffle Conjecture was stated. This refined conjecture was proved by E. Carlsson  Theorem 6.10 (The Shuffle Theorem). We have that where the sum is over all word parking functions of size n.
By applying the ζ-map of J. Haglund [Hag07] and interpreting the result, we obtain the equivalent statement now using vertical-strip LLT polynomials as they are defined in this paper. We have that where P * is the Schröder path obtained from the Dyck path P by replacing every instance of a pattern en by a diagonal step d. By applying Corollary 2.10 we obtain a new combinatorial formula for F n (x; q + 1, t).
We also note that vertical-strip LLT polynomials appear in a similar manner in the rational shuffle conjecture [BGLX15] and its subsequent proof [Mel16]. Similar observations can be used to show that the conjectured expressions for ∆ e k e n appearing in the Delta Conjecture by J. Haglund, J. Remmel and A. Wilson [HRW18] are also e-positive after the substitution q → q + 1.
We also note that the expression in [BCP18, Thm. 3.2.6], where we have r sets of variables (compared to the one or two sets of variables in the main literature of diagonal harmonics). In particular, an open question, [BCP18, Conj. 3.2.17], seem to indicate a close connection with vertical-strip LLT polynomials, due to an e-positivity phenomenon.
Another interesting result is the combinatorial interpretation of (−1) n−1 ∇p n . The Square Path Conjecture was formulated by N. Loehr and G. Warrington [LW07]. Later E. Sergel [Ser17] gave a proof of the Square Path Conjecture by using the Compositional Shuffle Theorem.
Theorem 6.12 (E. Sergel, (2017)). We have the expansion into fundamental quasisymmetric functions where the sum is taken over all preference functions of size n, and σ(w) is the reading word of w, and dinv is a statistic closely related to the statistic asc on colorings.
There is again a close connection with vertical-strip LLT polynomials. We shall give an example explaining the connection.
Example 6.13. A preference function is simply a map f : [n] → [n]. We can describe such an f by writing f −1 (i) in column i in an n × n-diagram, sorted increasingly from the bottom. Moreover, we demand that the entries f −1 (i) appear in rows lower than f −1 (j) for all i < j. This uniquely determines the diagram. The numbers in the digram are called cars 2 . There is a unique North-East path from (0, 0) to (n, n) where each North step is immediately to the left of a car. We set α i := |f −1 (i)|. For example, the preference function f = (5, 2, 5, 2, 2, 3) give rise to the diagram (a) here: The path P is given by ennneneennee. Moreover, α = (0, 3, 1, 0, 2, 0), and note that P is uniquely determined by α. There are a few other statistics on preference functions that only depend on α. We let area(α) be the number of squares below P and strictly above the lowest diagonal of the form y = x + k, containing a car. In our diagram above, area(α) = 8. Moreover, we let below(α) be the number of cars strictly below the y = x line. In this case, below(α) = 1, as the car labeled 2 is below the main diagonal.
Finally, given α, we associate a Schröder path P (α) as follows. We read the cars diagonal by diagonal as in (b), this gives the reading order of the cars. Then → xy is a diagonal steps in the Schröder path if and only if label x is immediately below y in n (−1) n−1 ∇p n 1 s 1 2 s 2 + (q + t + qt)s 11 3 s 3 + (q + q 2 + t + qt + q 2 t + t 2 + qt 2 + q 2 t 2 )s 21 + +(q 3 + qt + q 2 t + q 3 t + qt 2 + q 2 t 2 + q 3 t 2 + t 3 + qt 3 + q 2 t 3 )s 111 Table 1. The Schur expansion of (−1) n−1 ∇p n for the first few n.
the reading word order (b). We note that the map that takes P to the Schröder path is also a type of ζ-map, see [AHJ14, Sec. 5.2].
We can now state the expansion of (−1) n−1 ∇p n in terms of vertical-strip LLT polynomials. It is straightforward to deduce this from Theorem 6.12, by unraveling the definitions.
Theorem 6.14. We have the expansion where the sum is taken over all weak compositions α with n parts, of size n, and P (α) is as in Example 6.13.
By applying Corollary 2.10 we again obtain a new combinatorial expression for (−1) n−1 ∇p n and note that it is e-positive if we replace q with q + 1. We also mention [Ber13] and [Ber17,Sec. 4], where related e-positivity conjectures are made.
6.5. The Dyck path algebra. The Dyck path algebra A q is defined by E. Carlsson and A. Mellit in [CM17, Def. 5.1] as the path algebra over Q(q) of the quiver with vertex set N with directed edges d + from k to k + 1, directed edges d − from k + 1 to k, and loops T 1 , . . . , T k−1 at k for all k ∈ N, subject to the relations Here k denotes the vertex on which the respective paths begin, and we denote ϕ := (d − d + − d + d − )/(q − 1). The Dyck path algebra can be realised as an algebra of operators acting on (generalized) symmetric functions. For k ∈ N let V k := Λ ⊗ Q[y 1 , . . . , y k ], such that V 0 = Λ, and set For k ∈ N and i ∈ [k − 1] define T i : V k → V k by T i (f (x, q, y)) := (q − 1)y i+1 f (. . . y i , y i+1 . . . ) + (y i+1 − qy i )f (. . . y i+1 , y i . . . ) y i+1 − y i . For all k ∈ N define a raising operator d + : V k → V k+1 and a lowering operator d − : V k+1 → V k by d + (f (x; q; y)) := T 1 T 2 · · · T k (f [x + (q + 1)y k+1 ; q; y]), Here brackets denote plethystic substitution in the x-variables, and g| y i k denotes the coefficient of y i k in g. Finally for all k ≥ 1 define ϕ : V k → V k by To each Schröder path P = s 1 s 2 · · · s we associate an operator for all i ∈ [ ]. One of the fundamental insights of E. Carlsson and A. Mellit is that these path operators are creation operators of vertical-strip LLT polynomials.
In [CM17, Sec. 4] Theorem 6.15 is proven by analyzing the effect of the raising and lowering operators on characteristic functions of partial Dyck paths. This can be compared to our proof of the bounce relations in Section 4, which has the advantage that we do not need to introduce additional variables. The following theorem is a consequence of the results of M. D'Adderio in [D'A20].
In fact, this is precisely how Theorem 2.1 was discovered. M. D'Adderio proceeds to use Theorem 6.16 in conjunction with Theorem 6.15 to prove the e-positivity of vertical-strip LLT polynomials. On the other hand, combining Theorem 6.16 with Theorems 2.1 and 2.5 one obtains an alternative proof of Theorem 6.15. 6.6. Chromatic quasisymmetric functions. Chromatic symmetric functions were defined by R. Stanley as a symmetric function generalization of the chromatic polynomial of a graph in [Sta95]. These functions have received much recent attention since J. Shareshian and M. Wachs introduced a quasisymmetric q-analog in [SW16]. E. Carlsson and A. Mellit observed a plethystic relationship between chromatic quasisymmetric functions X P (x; q) of unit-interval graphs and unicellular LLT polynomials. This was implicitly done already in [HHL05a, Sec. 5.1]. It is also hidden in [GP16,Eq. (178)]. The relationship is as follows: Lemma 6.17 ([CM17, Prop. 3.5]). Let P be a Dyck path of size n. Then X P (x; q) = (q − 1) −n G P [x(q − 1); q], where the bracket denotes a substitution using plethysm.
Using Lemma 6.17 and Theorem 3.5 we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.
Corollary 6.18. The function that assigns to each Dyck path P the chromatic quasisymmetric function of the unit-interval graph Γ P is the unique function F : D → Λ, P → F P that satisfies the following conditions: (i) For all k ∈ N the initial condition F n(ne) k e = X n(ne) k e holds, where X n(ne) k e denotes the chromatic quasisymmetric function of the path on k + 1 vertices 3 . (ii) The function F is multiplicative, that is, F P Q = F P F Q for all P, Q ∈ D.
(vi) Let P ∈ D be a Dyck path and let (x, z) ∈ Z 2 be a point on P with x + 1 < z.
If the bounce decomposition of P at (x, z) is given by P = U n.nV ne.eW then F U n.nV ne.eW = (q + 1)F U n.nV e.neW − qF U n.nV e.enW .
If the bounce decomposition of P at (x, z) is given by P = U ne.nV ne.eW then F U en.nV en.eW + F U ne.nV ne.eW + F U n.neV e.enW = F U en.nV ne.eW + F U ne.nV e.enW + F U n.neV e.neW .
It is a major open problem to prove that the chromatic quasisymmetric functions X P (x; q) expand positively in the e-basis. This is known as the Shareshian-Wachs conjecture [SW12,Conj. 4.9], and for q = 1 it reduces to the Stanley-Stembridge conjecture [SS93,Sta95]. Some special cases have been solved, for example in [CH19, HP19, HNY20, FHM19, Dv18, Dah18, DFvW20] using different methods. Theorem 2.9 together with Lemma 6.17 implies the following expansion.
Why the right-hand side in (48) is e-positive remains very much unclear. We remark that vertical-strip LLT polynomials do not seem to have a natural pendent in the world of chromatic symmetric functions in general. In principle one could use (47) also when P is a Schröder path, however, the resulting functions are not even monomial positive. 6.7. Hessenberg varieties. Hessenberg varieties have been studied since the work of F. De Mari, C. Procesi, and M. Shayman [MPS92]. Following up a conjecture of J. Shareshian and M. Wachs, recently unicellular LLT polynomials and chromatic quasisymmetric functions of unit-interval graphs have been connected to the equivariant cohomology rings of regular semisimple Hessenberg varieties by P. Brosnan and T. Chow, and independently by M. Guay-Paquet [BC18,GP16]. Here we follow more or less the notation of [GP16].
A flag in C n is a sequence F • := (F 1 , F 2 , . . . , F n ) of subspaces {0} ⊂ F 1 ⊂ F 2 ⊂ · · · ⊂ F n = C n such that dim F i = i. The full flag variety Flag(C n ) is defined as the set of all flags in C n . Each Dyck path P ∈ D n is readily identified with a Hessenberg function, that is, a function h : [n] → [n] such that h(i) ≥ i for all i ∈ [n], and h(i + 1) ≥ h(i) for all i ∈ [n − 1]. Let M ∈ GL(n, C) be a diagonal matrix with n distinct eigenvalues.
The bounce decomposition of P at (x, z) is P = U n.nV d.eW where U = nne, V = nen and W = neeee. Therefore we need to consult Table 2 below. We have v = 1, w = 2, y = 4, Q = U n.nV e.dW , and importantly hrv(θ, x) = 8, hrv(θ, y) = 7, and hrv(θ, z) = 7. Thus we need to consider the weak standardisation of 877 which is 211. Table 2 shows that there are four subcases in the row indexed by σ = 211.
That is, all four orientations of the edges xy and yz could lead to this word. In our case → xy, → zy ∈ θ. Thus we are in the second subcase, which belongs to 5 . This means we should apply the map ψ of Lemma 5.14 and end up in case 5 in the column corresponding to Q. In accordance with what the table predicts, the resulting orientation θ contains → yx, and the weak standardization of hrv(θ , x), hrv(θ , y), hrv(θ , z) is 121. Since θ has one more descent than θ we should regard θ as the second subcase of 5 (in column Q), which is multiplied by q. The reader can check that if ψ is applied to an orientation from the first subcase of 5 (in column P ), then the number of ascents is preserved. Table 2. Subcases for orientations, first bounce relation. The missing σ ∈ {112, 123, 213} are not possible, since the edge → xz is present in P and → yz is present in Q. The maps marked with i are covered by Lemma 5.16, while i and i are covered by Lemma 5.12 and Lemma 5.8, respectively. Table 3. Subcases for orientations, second bounce relation. The maps marked with i correspond to Lemma 5.21 and the maps marked with i are covered by Lemma 5.25. The missing σ ∈ {112, 123, 213} are not possible, due to the edges → xz and → yz being present in P and Q, respectively.