Alessandro Iraci
email: alessandro.iraci@unipi.it

Philippe Nadeau
email: nadeau@math.univ-lyon1.fr

Anna Vanden Wyngaerd
email: anna.vanden.wyngaerd@ulb.be

Smirnov words and the Delta Conjectures

Keywords: Delta conjecture, coinvariant ring, Smirnov

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Introduction

This work is mainly concerned with a combinatorial expansion and its consequences. It is motivated by a circle of problems in representation theory, which we briefly survey in this introduction.

In the 1990s, Garsia and Haiman introduced the ring of diagonal coinvariants DR n . The study of the structure of this S n -module and its generalizations has been an important research topic in algebra and combinatorics ever since. The ring is defined as follows: consider the space C[x n , y n ] := C[x 1 , . . . , x n , y 1 , . . . , y n ] and define an S n -action as σ • f (x 1 , . . . , x n , y 1 , . . . , y n ) := f (x σ(1) , . . . , x σ(n) , y σ(1) , . . . , y σ(n) ) for all f ∈ C[x n , y n ] and σ ∈ S n . Let I(x n , y n ) be the ideal generated by the S n -invariants with vanishing constant term. Then the ring of diagonal coinvariants is defined as

DR n := C[x n , y n ]/I(x n , y n ).
The space has a natural bi-grading: let DR (i,j) n be the component of DR n with homogeneous x-degree i and homogeneous y-degree j. This grading is preserved by the S n -action. Garsia and Haiman conjectured, and Haiman later proved [START_REF] Haiman | Vanishing theorems and character formulas for the Hilbert scheme of points in the plane[END_REF], a formula for the graded Frobenius characteristic of the diagonal harmonics: grFrob(DR n ; q, t) := ∑ i,j∈N q i t j Frob(DR

(i,j) n ) = ∇e n , (1.1)
where e n is the n-th elementary symmetric function and ∇ is the operator introduced in [START_REF] Bergeron | Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions[END_REF]. In [START_REF] Haglund | A combinatorial formula for the character of the diagonal coinvariants[END_REF], the authors gave a combinatorial formula for this graded Frobenius character ∇e n , in terms of labelled Dyck paths, called the shuffle conjecture. It is now a theorem by Carlsson and Mellit [START_REF] Carlsson | A proof of the shuffle conjecture[END_REF]. The Delta conjecture is a pair of combinatorial formulas for the symmetric function ∆ ′ e n-k-1 e n in terms of decorated labelled Dyck paths, stated in [START_REF] Haglund | The Delta Conjecture[END_REF] -we detail the combinatorics in Section 5. Here ∆ ′ e n-k-1 is a certain symmetric function operator (depending on q, t). These conjectures reduce to the shuffle theorem when k = 0. This extension of the combinatorial setting led Zabrocki, D'Adderio, Iraci and Vanden Wyngaerd to introduce extensions of DR n [START_REF] Zabrocki | A module for the Delta conjecture[END_REF][START_REF] D'adderio | Theta operators, refined Delta conjectures, and coinvariants[END_REF]. Consider the ring C[x n , y n , θ n , ξ n ] where the x n , y n are the usual commuting (or bosonic) variables, while the θ n , ξ n are anti-commuting (or fermionic): θ i θ j = -θ j θ i and ξ i ξ j = -ξ j ξ i for all 1 ≤ i, j ≤ n.

Again, consider the S n -action that permutes all the variables simultaneously. If I(x n , y n , θ n , ξ n ) now denotes the ideal generated by the S n -invariants without constant term, define TDR n := C[x n , y n , θ n , ξ n ]/I(x n , y n , θ n , ξ n ). This ring is naturally quadruply graded: let TDR (i,j,k,l) n denote the component of TDR n of homogeneous (i, j, k, l)-degrees. In [START_REF] Zabrocki | A module for the Delta conjecture[END_REF] Zabrocki conjectured ∑ i,j∈N q i t j Frob(TDR

(i,j,k,0) n ) ? = ∆ ′ e n-k-1 e n . (1.2)
Note that the symmetric function of the Delta conjectures occurs on the right-hand side.

In [START_REF] D'adderio | Theta operators, refined Delta conjectures, and coinvariants[END_REF], D'Adderio with the first and third named authors introduced operators Θ f (depending on q, t), for any symmetric function f , and showed that ∆ ′ n-k-1 e n = Θ e k ∇e n-k . This permitted them to extend Zabrocki's conjecture as follows: ∑ i,j∈N q i t j Frob(TDR

(i,j,k,l) n ) ? = Θ e l Θ e k ∇e n-k-l .
(1.3) Special cases of the conjecture have been studied over the years. Let us call the "(a, b)-case" the structures linked to the diagonal coinvariant ring with a sets of bosonic variables and b sets of fermionic variables. The (2, 1)-and the (2, 2)-cases thus occur in (1.2) and (1.3) respectively, and the (2, 0)-case is the known case (1.1). The (1, 0) and (0, 1) cases are classical rings and the conjecture is known to hold in this case. The (1, 1)-case, or the superspace coinvariant ring, is still open, but Rhoades and Wilson in [START_REF] Rhoades | The Hilbert series of the superspace coinvariant ring[END_REF] showed that its Hilbert series agrees with the expected formula. The (0, 2)-case, or fermionic Theta case, was proved by Iraci, Rhoades, and Romero in [START_REF] Iraci | A proof of the fermionic theta coinvariant conjecture[END_REF].

In this abstract, we will turn our interest to the combinatorics that (conjecturally) occur in the (1, 2)-case. Following Conjecture (1.3), we thus are led to study the symmetric function Θ e k Θ e l ∇e n-k-l | t=0 .

Our combinatorial model is that of segmented Smirnov words. A Smirnov word is a word in the alphabet of positive integers such that adjacent letters are distinct. A segmented Smirnov word is the concatenation of Smirnov words with prescribed lengths (see Definition 2.1). The main result of this paper (Theorem 2.5) is an expansion in terms of segmented Smirnov words.

Theorem. For any n, k, l, we have the identity between symmetric functions in (x i ) i≥1

Θ e k Θ e l ∇e n-k-l | t=0 = ∑ w∈SW(n,k,l) q sminv(w) x w 1 x w 2 • • • x w n .
Here SW(n, k, l) is the set of segmented Smirnov words with k descents and l ascents, while the power of q is given by a new sminversion statistic on these words (see Definition 2.3). This expansion can be expressed more compactly in terms of fundamental quasisymmetric functions (Proposition 2.7).

The proof of the main theorem relies on an algebraic recursion (Proposition 2.4) for the symmetric function under study. We show in Section 3 that the combinatorial expansion satisfies indeed the same recursion.

In Section 4, we focus on the special case k + l = n -1 which turns out to be linked to various topics in the literature. In Section 5, we describe an explicit bijection between segmented Smirnov words and "doubly decorated labelled Dyck paths" (Theorem 5.1), motivated by a potential unified Delta conjecture.

Preliminaries and main result

Combinatorics. In this work Z + is the set of positive integers, and we will fix n ∈ Z + . We write µ ⊨ 0 n if µ is a weak composition of n, that is µ = (µ 1 , µ 2 , . . .) where the µ i are nonnegative integers that sum to n. A composition α ⊨ n is a finite sequence α = (α 1 , . . . , α t ) of positive integers that sums to n. Definition 2.1. A Smirnov word of length n is an element w ∈ Z n + such that w i ̸ = w i+1 for all 1 ≤ i < n. A segmented Smirnov word is a word w ∈ Z n + together with a composition α = (α 1 , . . . , α t ) ⊨ n such that if w is written as the concatenation w 1 • • • w t where each w i has length α i , then each w i is a Smirnov word.

Let SW(n) be the set of segmented Smirnov words of length n. We say that α is the shape of w. We call w 1 , . . . , w t segments of w. We usually simply denote a segmented Smirnov word by w, and omit the shape α. In examples, we separate segments by vertical bars. Segmented Smirnov words of shape (n) are naturally identified with Smirnow words of length n.

Given µ ⊨ 0 n, we denote by SW(µ) the set of segmented Smirnov words with content µ, that is they contain µ 1 occurrences of 1, µ 2 occurrences of 2, and so on. We clearly have SW(n) = µ⊨ 0 n SW(µ). We call segmented permutation a segmented Smirnov word in SW(1 n ). Note that these can be identified with pairs (σ, α) with σ ∈ S n and α ⊨ n.

Example 2.2. If µ = (2, 1), then SW(µ) has 8 elements: 1|1|2, 1|2|1, 2|1|1 with shape (1, 1, 1); 1|12, 1|21 with shape (1, 2); 21|1, 12|1 with shape (2, 1) and 121 with shape (3).

Given a Smirnov word w, we say that i is an ascent of w if w i+1 > w i , and a descent otherwise. If w ∈ SW(n), we say that i is an ascent (resp. descent) of w if it is an ascent (resp. descent) of one of its segments. Let us denote by SW(n, k, l) the set of segmented Smirnov words with k descents and l ascents; note that these words have nkl segments. For µ ⊨ 0 n, we also define SW(µ, k, l) as the intersection SW(µ) ∩ SW(n, k, l).

We can now define the main new statistic of this work. An index i ∈ {1, . . . , n} is called initial (resp. final) if it corresponds to the first (resp. last) position of a segment, i.e. if it has the form

i = α 1 + • • • + α m-1 + 1 (resp. i = α 1 + • • • + α m ) for some t ∈ {1, . . . , t}.
Definition 2.3 (The sminv statistic). For a segmented Smirnov word w of shape α ⊨ n, we say that (i, j) with 1 ≤ i < j ≤ n is a sminversion if w i > w j and one of the following holds:

1. j is initial in w; 2. w j-1 > w i ; 3. i ̸ = j -1, w j-1 = w i , and j -1 is initial in w; 4. i ̸ = j -1 and w j-2 > w j-1 = w i .
We let sminv(w) be the number of sminversions of w. The segmented Smirnov word w = 321|2131, has sminv equal to 4, since [START_REF] Bergeron | Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions[END_REF][START_REF] Mellit | A proof of the compositional Delta conjecture[END_REF], [START_REF] Carlsson | A proof of the shuffle conjecture[END_REF][START_REF] D'adderio | Tiered Trees and Theta Operators[END_REF], [START_REF] Carlsson | A proof of the shuffle conjecture[END_REF][START_REF] Haglund | A combinatorial formula for the character of the diagonal coinvariants[END_REF] and (4, 7) are its sminversions. Finally, define

SW q (µ, k, l) = ∑ w∈SW(µ,k,l) q sminv(w) .
In view of Example 2.2, we can compute SW q ((2, 1), 0, 0) = 1 + q + q 2 ; SW q ((2, 1), 1, 0) = 1 + q; SW q ((2, 1), 1, 1) = 1; and SW q ((2, 1), 0, 1) = 1 + q.

Let us note two important cases where the statistic sminv simplifies:

• When w is a segmented permutation σ, only cases (1) and ( 2) occur.

• When w has shape (n), i.e. w is a Smirnov word, only cases (2) and (4).

Symmetric functions. We refer to [START_REF] Stanley | Cambridge Studies in Advanced Mathematics. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin[END_REF]Ch. 7] for undefined terminology. Consider the ring Λ of symmetric functions in (x i ) i∈Z + with coefficients in Q(q). Let us define

SF(n, k, l) := Θ e k Θ e l ∇e n-k-l | t=0 ∈ Λ (2.1)
to simplify notations. Here h ⊥ j is the operator dual to multiplication by h j , with respect to the standard duality on Λ given by ⟨h λ , m µ ⟩ = δ λ,µ .

The following proposition is the key to the combinatorial interpretation:

Proposition 2.4. For any n, k, l with k + l < n, SF(n, k, l) satisfies h ⊥ j SF(n, k, l) = j ∑ r=0 j ∑ a=0 j ∑ i=0 q ( r-i 2 ) q ( a-i 2 ) n -k -l j -r -a + i q n -k -l -(j -r -a) -1 i q × n -k -l -(j -r -a + i) r -i q n -k -l -(j -r -a + i) a -i q SF(n -j, k -r, l -a) for any j ≥ 1. Moreover SF(0, k, l) = δ k,0 δ l,0 and SF(n, k, l) = 0 if n < 0.
We omit the proof of this proposition in this abstract: it comes from the specialization t = 0 of [START_REF] Romero | New identities for theta operators[END_REF]Theorem 8.2], with some extra elementary computations.

Main result. Define

SW x;q (n, k, l) = ∑ µ⊨ 0 n SW q (µ, k, l)x µ = ∑ w∈SW(n,k,l) q sminv(w) x w , (2.2)
where x w = ∏ n i=1 x w i .

Theorem 2.5. For any n, k, l with k + l < n, we have the identity

SF(n, k, l) = SW x;q (n, k, l). (2.3)
Expansion into fundamental quasisymmetric functions. Let w be a segmented Smirnov word. For 1 ≤ i ≤ n, we say that i is thick if i is initial or w i-1 > w i , and thin otherwise.

Definition 2.6. Let σ be a segmented permutation of size n, and i ∈ {1, . . . , n}. Let j be such that σ j = σ i+1 . We say that i is splitting for σ if either of the following holds:

• i and j are in the same segment of σ, and |i -j| = 1;

• i is thick and j is thin;

• i and j are both thin and i < j;

• i and j are both thick and j < i.

Let Split(σ) = {1 ≤ i ≤ n -1 | i is splitting for σ}. For any subset S ⊆ [n -1], let Q S,n be the fundamental quasisymmetric function associated to S (see [START_REF] Stanley | Cambridge Studies in Advanced Mathematics. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin[END_REF]Sec. 7.19]). Proposition 2.7.

SW x;q (n, k, l) = ∑ σ∈SW(1 n ,k,l) q sminv(σ) Q Split(σ),n .
The proof relieson grouping terms in the right-hand side of (2.2) using a certain "reading order". We omit it in this abstract.

Proof of Theorem 2.5

The proof consists in showing that the series SW x;q (n, k, l) satisfies the relations encoded in Proposition 2.4.

In detail, fix µ ⊨ 0 n nonzero, let F µ be the coefficient of x µ in the power series SF(n, k, l), and let the last nonzero part of µ be µ m = j. Then by taking the inner product of SF(n, k, l) with h µ in Proposition 2.4, we obtain a recurrence for F µ . Theorem 2.5 then claims that SW q (µ, k, l) obeys the same recurrence. Explicitly, let µ -be equal to µ except that µ - m = 0, and let s := nkl be the number of segments, then one has to show:

SW q (µ, k, l) = j ∑ i=0 j ∑ r=i j ∑ a=i q ( r-i 2 ) s -(j -r -a + i) r -i q q ( a-i 2 ) s -(j -r -a + i) a -i q × s j -r -a + i q s -j + r + a -1 i q SW q (µ -, k -r, l -a). (3.1)
We will sketch a bijective proof below. Since it is quite technical, let us first give the simpler proof in the case µ = 1 n , which boils down to the following proposition: Proposition 3.1. For any n, k, l with k + l < n, the polynomials SW q (1 n , k, l) satisfy

SW q (1 n , k, l) = [n -k -l] q SW q (1 n-1 , k, l) + SW q (1 n-1 , k -1, l) + SW q (1 n-1 , k, l -1) + SW q (1 n-1 , k -1, l -1) .
Proof. Given a segmented permutation on n -1 elements, we want to insert n in all possible ways. It can be done in four different manners:

1. as a new singleton segment. This keeps the number of ascents and descents the same, and increases the number of segments by one; 2. at the beginning of a segment. This creates no ascent and one descent, and keeps the number of segments the same; 3. at the end of a segment. This creates one ascent and no descents, and keeps the number of segments the same;

4. as an element merging two adjacent segments

• • • w 1 |w 2 • • • → • • • w 1 nw 2 • • • .
This creates an ascent and a descent, and decreases the number of segments by one. Each of these insertions can be done in s different ways, if s is the number of segments in the final segmented permutation. Moreover, the construction is injective: if i is such that σ i = n for some σ ∈ SW(1 n ), then by looking whether i is initial and/or final, one knows which of the four types of insertion was performed.

From this one sees that the proposition holds at q = 1. As for sminversions, one checks that inserting n does not modify the number of those involving letters in {1, . . . , n -1}. Moreover, the value n is part of a sminversion with all initial letters to its right. In each case, this increases sminv by all possible amounts between 0 and s -1 = nkl -1. The recursion of Proposition 3.1 follows.

Sketch of the proof of (3.1). The idea is the same as in the standard case above. Starting with a word in w ∈ SW(µ -), we want to insert j occurrences of the letter m (larger than all letters of w) to create a word w ′ in SW(µ, k, l). As in the standard case, we distinguish if the occurrences of m are initial and/or final. The complication comes from inserting several occurrences of m.

Pick i, a, r ≥ 0 such that i ≤ a ≤ j and i ≤ r ≤ j. Then we insert successively:

• i is occurrences of m that are neither initial nor final (this is done by merging adjacent segments as in the standard case); • ri occurrences of m that initial but not final; • ai occurrences of m that are final but not initial; • and finally jra + i singletons equal to m. Note that the total number of occurrences of m is indeed j. Since we want s = nkl segments in the end, we must have s + i -(jra + i) = sj + r + a segments in w. Also, w must have kr descents and la ascents so that the final word has k descents and l ascents.

The claim is that the number of ways to insert m is given by the coefficient of SW q (µ -, kr, la) in (3.1) at q = 1: each of the four binomial coefficients can be whown to correspond naturally to one of the cases above. To complete the proof, one needs to check that then number of sminversions behaves as wanted. We omit the details in this abstract.

The maximal case

k + l = n -1
We focus in this section on various aspects of the case k + l = n -1 of Theorem 2.5. The combinatorial side now involves only Smirnov words. It is also conjecturally giving the graded Frobenius characteristic of the subspace of the (1, 2)-coinvariant space of maximum total degree in the fermionic variables ζ n , ξ n (cf. (1.3)).

Chromatic symmetric function interpretation. Given a graph

G = (V, E), a proper coloring is a function c : V → Z + such that {i, j} ∈ E =⇒ c(i) ̸ = c(j). If V = [n]
, a descent of a coloring is an edge {i, j} ∈ E such that i < j and c(i) > c(j). The chromatic quasisymmetric function of G is defined as

X G (x; q) = ∑ c : V→Z + c proper q des(c) ∏ v∈V x c(v) ,
where des(c) is the number of descents of c.

For the path graph 2) . . . c(n) is a Smirnov word of length n, and vice versa, if w is a Smirnov word of length n, then c(i) = w i is a proper coloring of G n . It follows from Theorem 2.5

G n = 1 -2 -• • • -n, if c is a proper coloring then c(1)c(
X G n (x; u) = n-1 ∑ k=0 u k Θ e k Θ e n-k-1 e 1 q=1,t=0
. This suggests also the existence of an extra q-grading on the cohomology of the permutahedral toric variety V n : indeed the graded Frobenius characteristic of that cohomology is known to be given by ωX G n (x; u), see [START_REF] Shareshian | Chromatic quasisymmetric functions and Hessenberg varieties[END_REF].

Parallelogram polyominoes.

A parallelogram polyomino of size m × n is a pair of northeast lattice paths on a m × n grid, such that first is always strictly above the second, except on the endpoints (0, 0) and (m, n). A labelling of a parallelogram polyomino is an assignment of positive integer labels to each cell that has a north step of the first path as its left border, or an east step of the second path as its bottom border, such that columns are strictly increasing bottom to top and rows are strictly decreasing left to right. In [START_REF] D'adderio | Tiered Trees and Theta Operators[END_REF] it is conjectured that Θ e m-1 Θ e n-1 e 1 enumerates labelled parallelogram polyominoes of size m × n with respect to two statistics, one of which is (a labelled version of) the area, and the other is unknown.

It is immediate to see that parallelogram polyominoes of size (nk) × (k + 1) and area 0 are again in bijection with Smirnov words of length n with k descents, and proper colorings of G n with k descents. Indeed, reading the labels of such a polyomino bottom to top, left to right, yields a Smirnov word of size n with k descents, and the correspondence is bijective. In particular, sminversions on Smirnov words define a statistic on this subfamily of parallelogram polyominoes, proving the conjectural identity and partially answering Problem 7.13 from [START_REF] D'adderio | Tiered Trees and Theta Operators[END_REF] in the case when the area is 0.

The case q = 0. Note that in this case, it is known [START_REF] Iraci | A proof of the fermionic theta coinvariant conjecture[END_REF] that the symmetric function in Theorem 2.5 is the Frobenius characteristic of the (0, 2)-case. It was also shown that the high-degree part of this module has a basis indexed by noncrossing partitions. In particular, this means that there is a bijection between segmented permutations with one segment (that is, permutations) with zero sminv, and noncrossing partitions. Lemma 4.1. Permutations with zero sminv are exactly 231-avoiding permutations, that is permutations σ with no i < j < k such that σ k < σ i < σ j .

Proof. Let σ be a permutation, and suppose that it has a 231 pattern, that is, that there

exist indices i < j < k such that σ k < σ i < σ j . Let m = min j < a ≤ k | σ m < σ i ; by definition, i < j ≤ m -1,
and σ m-1 > σ i , so (i, m) is a sminversion of σ. It follows that permutations with zero sminv are 231-avoiding permutations. Since a sminversion in a permutation corresponds to a 231 pattern, this concludes the proof.

Let π be a noncrossing partition, and let ϕ(π) be the permutation that, in one line notation, is written by listing the segments of π sorted by their smallest element, with the elements of each segment sorted in decreasing order. Let us call decreasing run of a permutation σ a maximal subsequence of consecutive decreasing entries of σ (in one line notation): then the segments of π correspond to the decreasing runs of ϕ(π). For instance, if π = {{1, 2, 5}, {3, 4}, {6, 8, 9}, {7}}, then ϕ(π) = 521439867.

The map ϕ defines a classical bijection between noncrossing partitions of size n with k + 1 segments and 231-avoiding permutations with k descents. This recovers known numerology about the (0, 2)-case. Remark 4.2. More generally, standard segmented permutations with zero sminv can be characterized as 231-avoiding permutations where letters of a segment are smaller all than letters of the segments to its right. These can be easily counted, and we recover the total dimension of the (0, 2)-coinvariant ring given by ( 2n+1 n ).

Connection with the Delta conjectures

Let us first note that we recover known combinatorial interpretations when setting k = 0 (resp. l = 0) in Theorem 2.5. Indeed this gives an expansion over segmented Smirnov words with no descents (resp. ascents), and these are easily identified with ordered multiset partitions [START_REF] Rhoades | Ordered set partition statistics and the Delta Conjecture[END_REF]. In each case, the sminv statistic can moreover be seen to be distributed as the inv statistic on ordered set partitions.

The two different Delta conjectures are as follows:

∆ ′ e n-k-1 e n = Θ e k ∇e n-k = ∑ D∈LD(n) * k q dinv(D) t area(D) x D (5.1) ? = ∑ D∈LD(n) •k q dinv(D) t area(D) x D .
(5.

2)

The sets LD(n) * k and LD(n) •k denote labelled Dyck paths of size n with k decorations on rises or valleys, respectively; and the statistics dinv and area depend on the decorations. So The rise version was recently proved in [START_REF] Mellit | A proof of the compositional Delta conjecture[END_REF].

Let us make some of the combinatorics explicit. A Dyck path of size n is a lattice path starting at (0, 0), ending at (n, n), using only unit North (N) and East (E) steps, and staying weakly above the line x = y. A labelled Dyck path is a Dyck path together with a positive integer label on each of its vertical steps such that labels on consecutive vertical steps must be strictly increasing (from bottom to top).

A rise of a labelled Dyck path is a North step that is preceded by another North step. A (contractible) valley of a labelled Dyck path is a vertical step v that is preceded either by two horizontal steps, or by a horizontal step that is preceded by a vertical step whose label is strictly smaller than v's label.

A decorated labelled Dyck path D is a labelled Dyck path, together with a choice of rises and (contractible) valleys, which are decorated. Let DRise(D), resp. DValley(D), be the set of i ∈ [n] such that the i-th vertical step of D is a decorated rise, resp. a decorated valley. We decorate rises with a * and valleys with a •. The set of decorated labelled Dyck paths with k decorated rises and l decorated valleys, is denoted by LD(n) * k,•l . The sets LD(n) * k , resp. and LD(n) •l , above correspond to setting l, resp. k, to 0. * * * * * * (5.3) Take D to be the left path in Figure 1. We have DRise(D) = {2, 6}, DValley = {3, 7}. Its area word of D is 01112320, and so its area equals 6.

The statistic dinv(D) counts "diagonal inversions" minus the number of decorated valleys; we omit its precise definition in this abstract.

In [START_REF] D'adderio | Theta operators, refined Delta conjectures, and coinvariants[END_REF], the authors conjectured a partial formula for a possible unified Delta conjecture, for which they have significant computational evidence:

Θ e k Θ e l ∇e n-k-l | q=1 ? = ∑ D∈LD(n) * k,•l t area(D) x D , ( 5.4)
The goal would thus be to find a statistic qstat : LD(n) * k,•l → N so that q qstat(D) t area(D) x D ;

(5.5)

and such that when k = 0 or l = 0, the formula reduces to (5.1) or (5.2), respectively.

Let us now come back to our setting. Comparing our main Theorem 2.5 at q = 1 with (5.4) at t = 0, we get the conjectural existence of a bijection between labelled Dyck paths of area zero and segmented Smirnov words. This bijection exists indeed: Let LD 0 (n) * k,•l be the subset of area zero Dyck paths in LD(n) * k,•l . Theorem 5.1. For any n, k, l, there is a bijection ϕ between SW(n, k, l) and LD 0 (n) * k,•l such that x w = x ϕ(w) . Sketch of the proof. Paths in LD 0 (n) * k,•l have a very specific shape: they are the concatenation of paths of the form N i E i , where all rises are decorated; see Figure 1, right. This precisely ensures that the area is zero, cf. Formula (5.3).

For µ ⊨ 0 n, and let LD 0 (µ) * k,•l be the subset of LD 0 (n) * k,•l such that x D = x µ . Using the special structure detailed above, one can then show bijectively that the cardinalities of the sets of LD 0 (µ) * k,•l decompose as SW q=1 (µ, k, l): namely, they satisfy (3.1) at q = 1. By matching with the bijective decomposition of SW(µ, k, l) in Section 3, we can obtain a recursively defined bijection ϕ between the two sets. We omit the details in this abstract.

What about q ? By transporting the sminv statistic through the bijection ϕ, we get a qstatistic on LD 0 (n) * k,•l . Now this statistic will not satisfy the unified Delta conjecture (5.5) at t = 0, because it does not match the dinv-statistic coming from the rise Delta conjecture.

It is however possible to fix this -thus we do have a unified Delta conjecture at t = 0by recursively defining a different q-statistic on SW(n). Roughly put, this is done by ordering segments in ad hoc ways when proving the recursion for SW(µ, k, l) (for sminv we simply order segments right to left). The drawback is that the statistic thus defined seems hard to characterize non recursively.

( 5 . 1 )

 51 is referred to as the rise version and (5.2) as the valley version of the Delta conjecture.

Figure 1 :

 1 Figure 1: Elements of LD(8) * 2,•2 (left) and LD 0 (8) * 4,•2 (right).

Θ

  e k Θ e l ∇e n-k-l ? = ∑ D∈LD(n) * k,•l

Partially supported by the project ANR19-CE48-011-01 (COMBINÉ).