Combinatorics of the Delta conjecture at q=-1

In the context of the shuffle theorem, many classical integer sequences appear with a natural refinement by two statistics $q$ and $t$: for example the Catalan and Schr\"oder numbers. In particular, the bigraded Hilbert series of diagonal harmonics is a $q,t$-analog of $(n+1)^{n-1}$ (and can be written in terms of symmetric functions via the nabla operator). The motivation for this work is the observation that at $q=-1$, this $q,t$-analog becomes a $t$-analog of Euler numbers, a famous integer sequence that counts alternating permutations. We prove this observation via a more general statement, that involves the Delta operator on symmetric functions (on one side), and new combinatorial statistics on permutations involving peaks and valleys (on the other side). An important tool are the schedule numbers of a parking function first introduced by Hicks; and expanded upon by Haglund and Sergel. Other empirical observation suggest that nonnegativity at $q=-1$ holds in far greater generality.


Introduction
In the early 2000's, Haglund, Haiman, Remmel, Loehr and Ulyanov stated the shuffle conjecture [10]: a combinatorial formula for the symmetric function ∇en in terms on labelled Dyck paths.The interest in the symmetric function ∇en (where ∇ is the MacDonald eigenoperator introduced in [1]) stems from it being the bi-graded Frobenius characteristic of the diagonal harmonic representation of the symmetric group [14].More than a decade after its statement, Carlsson and Mellit proved the full shuffle conjecture, which thus became a theorem [2].By then, many special cases were known: for example ∇en, en gives the famous q, t-Catalan numbers [8] and ∇en, h d e n−d the q, t-Schröder numbers [9].A consequence of the full shuffle theorem is that the bi-graded Hilbert series ∇en, h n 1 gives a q, t-analogue of (n + 1) n−1 .It can be described combinatorially as the generating function of length n parking functions with respect to area and number of diagonal inversions.
The famous Euler numbers (En) n≥0 can be defined by their generating series: They answer various enumeration problems, the most famous one being that En is the number of alternating permutations in Sn, that is, those σ such that σ1 > σ2 < σ3 > • • • .They also appear in Arnold's theory of singularity, and in number theory via their relation with Bernoulli numbers.Seeing them in the context of Macdonald q, t-combinatorics is new, and is the motivation for this project.We will show that specializing q = −1 in the q, t-analog of (n + 1) n−1 (the bi-graded Hilbert series of diagonal harmonics) gives: where En(t) is a t-analogue of En appearing in [15].This specialization at q = −1 is a trefinement of the identity P ∈PFn (−1) area(P ) = En, where PFn are the parking functions of size n.The history of this identity can be found in [18,21,22].For the definition of parking functions and their correspondence to standardly labelled Dyck paths, see [12,Chapter 5].
We will establish Equation (1) as a corollary of a more general statement involving a generalization of the shuffle theorem: the valley version of the Delta conjecture [11].This is a combinatorial formula for the symmetric function ∆ ′ e n−k−1 en.We will mainly use the following consequence of the Delta conjecture: q dinv(P ) t area(P ) x P , where stLD(n) •k denotes the set of standardly labelled Dyck paths with k decorated valleys and dinv and area are combinatorial statistics on this set.See Section 2.1 for the precise combinatorial definitions.At k = 0, we have ∆ ′ e n−1 en = ∇en, and the Delta conjecture reduces to the shuffle theorem.
Specializations of the shuffle theorem and Delta conjecture at q = 0 or q = 1 have been extensively studied (see [7] and [24], respectively).To our knowledge, apart from (2), nothing much was known about the specialization at q = −1.
We were inspired by the following remarkable symmetric function identity, which first appeared in [5,Theorem 4.11 Taking the scalar product with h n 1 and evaluating at q = −1, we obtain where the second equality is an easy consequence of the shuffle theorem.
Our main result is a combinatorial interpretation of the terms of this sum, conditional on (3).
Theorem 1.1.For all n ∈ N, we have where inv3 is a new statistic on permutations generalising Chebikin's notion of alternating descents [3] and monot(σ) is the number of double ascents or descents of σ (see Section 4 for the precise definitions).
Thus if the Delta conjecture is proven to be true, we will have the following symmetric function interpretation.
Notice that at z = 1 our theorem agrees with Equation (5).The specialisation at z = 0 of our theorem, and the fact that at k = 0 the Delta conjecture reduces to the shuffle theorem will imply our formula (1).
Our proof relies on the schedule formula decomposition of the combinatorial side of the valley Delta conjecture provided in [13].We use this schedule framework to identify the valley decorated Dyck paths that do not cancel out when specializing to q = −1.We then provide a bijection between these paths and permutations.This map will be defined via specific generating trees of the objects and will send area to inv3 and the number of decorations to monot.In this way the paths with no decorations (k = 0) get sent to permutations with no double ascents or descents, that is, alternating permutations.

Valley-decorated labelled Dyck paths
Definition 2.1.A Dyck path of size n is a lattice path going from (0, 0) to (n, n) consisting of east or north unit steps, always staying weakly above the line x = y, called the main diagonal.The set of dyck paths is denoted by D(n).Definition 2.2.A labelled Dyck path is a pair (π, w), where π ∈ D(n) and w its labelling: a word of positive integers whose i-th letter labels the i-th vertical step of π, placed in the square to the right of this step, such that the labels appearing in the same column are increasing from bottom to top.A labelling is said to be standard if its the labels are exactly 1, 2, . . ., n.The set of (standardly) labelled Dyck paths of size n is denoted by LD(n) (respectively, stLD(n)).
Standardly labelled Dyck paths are in bijection with parking functions.
Definition 2.3.The area word of a Dyck path π ∈ D(n) is the word a of n non-negative integers whose i-th letter is the number of whole squares between the i-th vertical step of π and the main diagonal x = y.The area of a Dyck path is the sum of the letters of its area word and is denoted by area(π).Definition 2.4.Given P := (π, w) ∈ LD(n) with area word a, the i-th vertical step of P is called a contractible valley if • or ai−1 = ai and wi−1 < wi.In other words, the i-th vertical step is a contractible valley if it is preceded by a horizontal step and the following holds: after replacing the two steps with (and accordingly shifting the i-th label one cell to the left), we still get a valid labelled path where labels are increasing in each column.Definition 2.5.A (valley) decorated labelled Dyck path is a triple (π, w, dv) where (π, w) ∈ LD(n) and dv is some subset of the contractible valleys of (π, w).The elements of dv are called decorations and we visualise them by drawing a • to the left of these contractible valleys.The set LD(n) •k denotes the decorated labelled Dyck paths with exactly k decorations.Definition 2.6.Given P := (π, w, dv) ∈ LD(n) •k with area word a, a pair (i, j) of indices of vertical steps with 1 < j ≤ n is said to be a • primary diagonal inversion if ai = aj, wi < wj and i ∈ dv, • secondary diagonal inversion if ai = aj + 1, wi > wj and i ∈ dv.The dinv of P is defined to be the total number of primary and secondary dinv pairs minus the number of decorated valleys and is denoted by dinv(P ).
Remark 2.7.We note that the dinv of a decorated labelled path is always a non-negative integer.Indeed, upon some reflection, one notices that each contractible valley forces the existence of at least one dinv pair.Definition 2.8.Given P := (π, w, dv) ∈ LD(n) •k , the area of P is simply defined as the area of the underlying Dyck path, disregarding the labels and decorations: area(P ) := area(π).
Thus, since there are 4 decorated valleys, the dinv is equal to 2.

Symmetric functions
For all the undefined notations and the unproven identities, we refer to [4], where definitions, proofs and/or references can be found.We denote by Λ the graded algebra of symmetric functions with coefficients in Q(q, t), and by , the Hall scalar product on Λ, defined by declaring that the Schur functions form an orthonormal basis.The standard bases of the symmetric functions are the monomial {m λ } λ , complete {h λ } λ , elementary {e λ } λ , power {p λ } λ and Schur {s λ } λ bases.
For a partition µ ⊢ n, we denote by Hµ := Hµ[X] = Hµ[X; q, t] = λ⊢n K λµ (q, t)s λ the (modified) Macdonald polynomials, where are the (modified) Kostka coefficients (see [12] for more details).Macdonald polynomials form a basis of the ring of symmetric functions Λ.This is a modification of the basis introduced by Macdonald [20].
If we identify the partition µ with its Ferrer diagram, i.e. with the collection of cells {(i, j) | 1 ≤ i ≤ µj , 1 ≤ j ≤ ℓ(µ)}, then for each cell c ∈ µ we define the co-arm and co-leg (denoted respectively as a ′ µ (c), l ′ µ (c)) as the number of cells in µ that are strictly to the left and below c in µ, respectively (see Figure 2).Define the following constant: Let f [g] denotes the plethystic evaluation of a symmetric function f in an expression g (see [12, Chapter 1 page 19]).

The statement
Now we have all the necessary definitions to state the valley Delta conjecture, which first appeared in [11].
Conjecture 2.12.For all n, k ∈ N, where the sum is over the set of labelled Dyck paths of size n with k decorations on contractible valleys.
Taking the (Hall) scalar product with h n 1 (i.e., the Hilbert series), the Delta conjecture implies (3).

Schedule formula
In this section we discuss the schedule formula for the combinatorics of the valley Delta conjecture proved by Haglund and Sergel in [13].Their formula is an extension of the first work on schedule numbers by Hicks in her thesis [16].σi.
n , we denote by dec(σ) its number of decorations.Definition 3.2.Given P := (π, w, dv) ∈ stLD(n) •k with area word a, the diagonal word of a decorated labelled Dyck path is the decorated permutation of n obtained as follows.A label wi of P is said to lie in the j-th diagonal if ai = j.List all the labels wi in the 0-th diagonal, in decreasing order, adding a decoration on the label if i ∈ dv.Then do the same for the 1-st diagonal, 2-nd diagonal, and so forth.Denote this diagonal word by dw(P ).
Example 3.3.The diagonal word of the path in Figure 1 is .8 .63 .274 .15. Definition 3.4.For σ ∈ Sn a permutation, its major index is defined to be the sum of the elements of the set {i ∈ [n − 1] | σ(i) > σ(i + 1)}.The reverse major index of a permutation is simply the major index of the reverse permutation σ rev := σn • • • σ1.For any marked permutation σ ∈ S • n , denote by revmaj(σ) the reverse major index of its underlying permutation.
The following is an easy consequence of the definitions.
Proof.Due to the condition of strictly increasing labels in the columns of labelled Dyck paths, each diagonal has at least one label which is bigger than some label of the previous diagonal.Thus, the labels in the j-th diagonal of P are exactly the numbers in the (j + 1)-th decreasing run of dw(P ), and they each contribute j units to the area.
• If τi is undecorated and an element of rj, let • If τi is decorated and an element of rj, let si =#{k ∈ rj | k is undecorated and k < τi} To reformulate this definition of schedule numbers, we introduce the following.Definition 3.8.Let σ ∈ Sn.A sequence of consecutive elements σi, . . ., σj (with 0 ≤ i ≤ j) in σ is a cyclic (decreasing) run if there exists an integer k such that σi + k mod n + 1, . . ., σj + k mod n + 1 is decreasing (where the modulo means we take the representative in {0, . . ., n}).Moreover, a cyclic run σi, . . ., σj is left-maximal if either i = 0 or σi−1, . . ., σj is not a cyclic run, and right-maximal if either j = n or σi, . . ., σj+1 is not a cyclic run.
Note that for each j, there is a unique left-maximal cyclic run σi, . . ., σj (obtained by choosing i to be minimal), and similarly for each i there is a unique right-maximal cyclic run σi, . . ., σj (obtained by choosing j to be maximal).Remark 3.10.The definition of schedules is rephrased as follows: if τi is undecorated (respectively, decorated), then si is the number of undecorated values (excluding τi) in the maximal decreasing cyclic run ending (respectively, starting) at τi. Example 3.11.If τ is the diagonal word of the path in Figure 1, then we have τ 0 .8 .6 3 .2 7 4 . 1 5 In Figure 3, we visualise τ by placing dots at coordinates (i, τi): white dots for undecorated and black dots for decorated τi.Notice that if we view this picture as a cylinder, identifying the top and the bottom, the operation σi → σi + k mod n can be seen as a rotation of this cylinder, hence the name "cyclic runs".The maximal cyclic run starting at .6 is .
27 (Figure 3, left), so that the schedule number is 2. The maximal cyclic run ending at 4 is 3 . 274 (Figure 3, right), so the schedule number is also 2.
A consequence of this theorem is that a marked permutation is the diagonal word of some permutation if and only if all its schedule numbers are strictly positive.At q = −1, more terms vanish as a consequence of the following.Proof.Suppose there is an index i such that si > 1.Our goal is to show that there exists i ′ such that s i ′ = si − 1.We distinguish two cases, (which are very similar): • σi is not decorated.Consider the left-maximal run ending at σi, denoted R, so R = (σ h , . . ., σi) for some h.Consider the maximal i ′ such that σ i ′ is not decorated, and h ≤ i ′ < i.It exists because si > 0. Let R ′ = σ h ′ , . . ., σ i ′ be the left-maximal cyclic run ending at σ i ′ .We have h ′ ≤ h (because we know that (σ h , . . ., σ i ′ ) is a cyclic run), so R and R ′ overlap on every non-decorated value of R other than σi.We deduce that s i ′ ≥ si − 1.In case of equality, we are done.Otherwise, we iterate this construction to find i ′′ < i ′ such that s i ′′ ≥ s i ′ − 1, etc. Since the increment is at most 1, after some steps we find i (k) such that s i (k) = si − 1.
• σi is decorated.Consider the right-maximal run beginning at σi, denoted R, so R = (σi, . . ., σj) for some j.Consider the maximal i ′ such that σ i ′ is not decorated, and i < i ′ ≤ j.It exists because si > 0. Let R ′ = σ h ′ , . . ., σ i ′ be the left-maximal cyclic run ending at σ i ′ .We have h ′ ≤ i (because we know that (σi, . . ., σ i ′ ) is a cyclic run), so R and R ′ overlap on every non-decorated value of R other than σ i ′ .We deduce that s i ′ ≥ si − 1.In case of equality, we are done.Otherwise, we are back to the situation in the previous case.
In either case, we eventually find i ′ as announced.The statement in the lemma follow by iteration.
A consequence of this lemma is that the product n i=1 [si]q at q = −1 is 0 unless all schedule numbers are 1 (otherwise, one of the factors is [2]q = 1 + q).Therefore, the schedule formula of Theorem 3.12 becomes (−1) dinv(P ) t area(P ) = t revmaj(τ ) .
Notation.We denote by S • n (1 n ) to be the subset of S • n of marked permutations with schedule 1 n .Now, summing Equation ( 6) over all possible permutations with k decorations τ , we get the following interpretation of the combinatorics of the Hilbert series of the Delta conjecture at q = −1: Remark 3.14.We announced in the introduction that in the case k = 0 and t = 1, the lefthand side of ( 7 We close this section by providing a bijection between Sn and S • n (1 n ), whose inverse is given by simply removing the decorations.The existence of this bijection means that each permutation can be decorated in exactly one way, so that the result has schedule 1 n .Lemma 3.15.Let σ ∈ S • n (1 n ).Let σ k , . . ., σ ℓ be a left-maximal cyclic run, and assume that σ ℓ is undecorated.Then σ k is undecorated as well, and σ k+1 , . . ., σ ℓ−1 are decorated.Proof.Since s ℓ = 1, there is exactly one undecorated entry in σ k , . . ., σ ℓ−1 .Denote σu this undecorated entry.If u = k, σ k is decorated and has at least two decorated entries (namely σu and σ ℓ ) in the right-maximal cyclic run beginning at σ k .This would give s k ≥ 2, which contradicts s k = 1.Thus u = k, and we get that σ k in the only non-decorated entry in σ k , . . ., σ ℓ−1 .Lemma 3.16.For each permutation σ ∈ Sn, there exists exactly one decorated permutation with underlying permutation σ and schedule 1 n .Proof.Assuming that there exists a decoration of σ so that the schedule is 1 n , the previous lemma readily gives necessary conditions on how to find it (in particular, uniqueness will follow from existence): we proceed from right to left (starting from σn and ending at σ1), noting than σn is necessarily undecorated (otherwise we have sn = 0).We define a sequence of indices i1 > i2 > • • • > im for some m ≥ 1 as follows; • knowing ij , we find ij+1 by the condition that σi j+1 • • • σi j is the left-maximal cyclic run ending at σi j , • the sequence stops at im = 0.
We claim that decorating the indices not in {i1, . . ., im} yields the unique decoration such that the associated schedule is 1 n .It remains only to check that the schedule of this decorated permutation is indeed 1 n .By construction, we have s k = 1 if σ k is undecorated.It remains to show s k = 1 when σ k is decorated.So, consider a right-maximal cyclic run σ k , . . ., σ ℓ where σ k is decorated.
• Suppose s k ≥ 2. So, there are two (or more) undecorated entries in this run, say σi and σj with k < i < j.The left-maximal cyclic run ending at σj contains at least σ k , . . ., σj, so σi being undecorated contradicts the construction of i1 > i2 > • • • > im as above.
• Suppose s k = 0. So, there are no undecorated entries in this run.Let σi be an undecorated entry with i > k and i minimal (this exists because σn is undecorated).We have i > ℓ (because σ k+1 , . . ., σ ℓ are decorated).Consider the left-maximal cyclic run ending at σi, denoted σ i ′ , . . ., σi.It cannot begin at σ i ′ with i ′ ≤ k, because σ k , . . ., σi is not a cyclic run (the right-maximal cyclic run beginning at σ k ends at σ ℓ , and i > ℓ).Thus i ′ > k, so σ i ′ , . . ., σi does not contain any undecorated entry apart from σi.But this means si = 0, which is a contradiction.
Other cases being excluded, we thus have s k = 1.This completes the proof of existence and uniqueness of the decoration with schedule 1 n .

Permutations
We continue to use Convention 3.6: for any σ ∈ Sn, we set σ0 = 0.
Definition 4.3.Let σ ∈ Sn, a pair (i, j) with 1 ≤ i < j ≤ n is said to be a 3-inversion if one of the following holds: • σj is a double ascent and σj−1 < σi < σj ; • σj is a double descent and σj−1 > σi > σj; • σj is a peak and σi > σj; • σj is a valley and σi < σj.The number of 3-inversions of σ is denoted by inv3(σ).Though the definition of the statistic might not seem very natural, we will see in the proof of the proposition that it can be tracked via a rather simple insertion procedure on permutations (similar to the Lehmer codes).
We recall the following classical definitions.Two classical Mahonian statistics are the major index (Definition 3.4) and the inversion number defined by inv(σ) = # (i, j) 1 ≤ i < j ≤ n and σi > σj .From the main result in Section 5 (Theorem 5.1) and the fact that revmaj is Mahonian, we will be able to deduce the following.

En
x n n! .
In [15], the authors introduced an interesting t-analogue to the Euler numbers that was subsequently studied in [3] and [17].Definition 4.12.For all n ∈ N, define: This polynomial En(t) has several beautiful properties including the facts that the generating functions n≥0 E2n(t)z n and n≥0 E2n+1(t)z n have nice continued fraction expressions [15,17].
Here we study a shift of this t-analogue, namely t ⌊n 2 /4⌋ En(t).This t-analogue is naturally connected to our 3-inversion statistics.
Finally, the following property follows easily from Definitions 5.4 and 5.5.

A second tree related to peaks and valleys
Definition 5.11.Define a tree T2 of permutations as follows.Take its root to be 1 ∈ S1.For σ ∈ Sn and 1 ≤ l ≤ n + 1, define η l (σ) ∈ Sn+1 to be the unique permutation σ ′ such that σ ′ n+1 = l and σ ′ 1 , . . ., σ ′ n are in the same relative order as σ1, . . ., σn.In other words for 1 ≤ l ≤ n + 1.This permutation is called the insertion of l in σ.The η l (σ) will form the descendants of σ in T2.
See the tree on the right in Figure 5. Later (Definition 5.14), we will define a total order on the descendants of a node in T2, in a way that will give the isomorphism with T1.This ordering will closely depend upon the following quantity.Definition 5.12.Given σ ∈ Sn, define it structural attribute: where we consider σ0 = 0 in case n = 1.
Proof.Take σ ∈ Sn and set σ ′ = η l (σ).Notice that for j ≤ n we have that (i, j) forms a 3-inversion in σ if and only if it forms a 3 inversion in σ ′ (see Definitions 4.3 and 5.11).It follows that inv3(σ ′ ) = inv3(σ ′ ) + s where s is the number of 3-inversions of the form (i, n + 1).We distinguish two cases.We have drawn a schematic representation of the proof in Figure 4.
1. Either σ ends with an ascent, that is σn−1 < σn.So we have ã := ã(σ) = n + 1 − σn.This is illustrated in the left part of Figure 4, where the values in red are the value of ψ(l).
Furthermore, we have:

Isomorphism of the trees
To prove Theorem 5.1, we define φ recursively as follows: In Figure 5, the k-th descendent of each node is the k-th descendant from the bottom, so that the image by φ of an element in the left tree can be obtained by looking for the corresponding element in the right tree, if we were to "superpose" one tree on the other.
It would thus be interesting to study the q = −1 evaluation in a more general framework; for example in modified Macdonald polynomials.
It would be very interesting to give a combinatorial proof of (11).Of course, one might try do to this starting from the other combinatorial interpretation: dec(σ)≤j q revmaj(σ) .
In the case t = 1, it is possible to give a combinatorial proof of the conjecture, based on the combinatorial interpretation in (10).The idea is to consider the map σ → σ ′ (where, for σ ∈ Sn, σ ′ ∈ Sn−1 is obtained by removing the entry n), and examine how monot is distributed among the n pre-images of a given σ ∈ Sn−1.

Figure 2 :
Figure 2: co-arm and co-leg of a cell in a partition.

Definition 3 . 1 .
The set of decorated permutations S •k n of [n] := {1, . . ., n} is the set of permutations of[n]  where k of its n letters are decorated, represented as .

n+1 1 σnFigure 4 :
Figure 4: Contribution to inv 3 for the n + 1 different insertions into a permutation of n.