A valley version of the Delta square conjecture
Alessandro Iraci Anna Vanden Wyngaerd
math.CO

	

Inspired by <cit.> and <cit.>, we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in <cit.> to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in <cit.>, relying on the formulation of the touching version in terms of the Θ_f operators introduced in <cit.>.

	
Intelligent Reflecting Surface Assisted Beam Index-Modulation for Millimeter Wave Communication
    Sarath Gopi and Sheetal Kalyani and Lajos Hanzo 	2020 IEEE.  Personal use of this material is permitted.  Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
    December 30, 2023
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§ INTRODUCTION

In <cit.>, Haglund, Remmel and Wilson conjectured a combinatorial formula for Δ_e_n-k-1'e_n in terms of decorated labelled Dyck paths, which they called Delta conjecture, after the so called delta operators Δ_f' introduced by Bergeron, Garsia, Haiman, and Tesler <cit.> for any symmetric function f. There are two versions of the conjecture, referred to as the rise and the valley version. 

In the same article <cit.> the authors conjecture a combinatorial formula for the more general expression Δ_h_mΔ_e_n-k-1'e_n in terms of decorated partially labelled Dyck paths, which we call generalised Delta conjecture (rise version). In this paper, the authors also state a touching refinement (where the number of times the Dyck path returns to the main diagonal is specified) of their conjecture. In <cit.>, the authors introduce the Θ_f operators, and reformulate the touching version using these tools. In the present work, we will be using the latter formulation.

The Delta conjecture and its derivatives have attracted considerable attention since their formulation, see among others Wilson-Equidistribution, Rhoades-2018, Remmel-Wilson-2015, Zabrocki-Delta-Module-2019, Haglund-Rhoades-Shimozono-Advances, Garsia-Haglund-Remmel-Yoo-2019, DAdderio-Iraci-VandenWyngaerd-GenDeltaSchroeder-2019, DAdderio-Iraci-VandenWyngaerd-TheBible-2019, DAdderio-Iraci-VandenWyngaerd-DeltaSquare-2019, DAdderio-Iraci-VandenWyngaerd-Delta-t0-2018, Zabrocki-4Catalan-2016, Qiu-Wilson-2019,Haglund-Sergel-2019. Most of the earlier work concerns the rise version, but interest in the valley version is growing. 

The special case k=0 of the Delta conjecture, which is known as the shuffle conjecture <cit.>, was recently proved by Carlsson and Mellit <cit.>. The shuffle theorem, thanks to the famous n! conjecture, now n! theorem of Haiman <cit.>, gives a combinatorial formula for the Frobenius characteristic of the 𝔖_n-module of diagonal harmonics studied by Garsia and Haiman.

In <cit.> Loehr and Warrington conjecture a combinatorial formula for Δ_e_nω(p_n)=∇ω(p_n) in terms of labelled square paths (ending east), called the square conjecture. The special case · ,e_n of this conjecture, known as the q,t-square, has been proved by Can and Loehr in <cit.>. Recently the full square conjecture has been proved by Sergel in <cit.>, who showed that the shuffle theorem by Carlsson and Mellit <cit.> implies the square conjecture (now square theorem).

In <cit.> the authors conjecture a combinatorial formula for [n-k]_t/[n]_tΔ_h_mΔ_e_n-kω(p_n) in terms of rise-decorated partially labelled square paths that we call generalised Delta square conjecture (rise version). This conjecture extends the square conjecture of Loehr and Warrington <cit.> (now a theorem <cit.>), i.e. it reduces to that one for m=k=0. Moreover, it extends the generalised Delta conjecture in the sense that on decorated partially labelled Dyck paths gives the same combinatorial statistics.

In <cit.>, the authors state a generalised Delta conjecture (valley version), extending the valley version of the Delta conjecture. They also prove the case q=0, extending the results in <cit.>. 

Inspired by <cit.> and <cit.>, we formulate two statements that can reasonably be called the generalised Delta square conjecture (valley version). One is a combinatorial interpretation of the symmetric function 
    [n-k]_q/[n]_qΔ_h_mΔ_e_n-kω(p_n)= [n]_t/[n-k]_tΔ_h_mΘ_e_k∇ω(p_n-k)
 (notice the swapping of q and t with respect to the rise version). The other is an interpretation of Δ_h_mΘ_e_k∇ω(p_n-k), for which the combinatorics seems to be nicer and does not have the multiplicative factor.  

Next, we adapt the schedule formula in <cit.> to objects with repeated labels, which enabled us to incorporate the monomials into the formula. This allowed us to obtain a schedule formula for the combinatorics of our conjecture and to deal with the symmetric functions more easily. As a byproduct, our formula provides a new factorisation of all other previous schedule formulae concerning Dyck or square paths.

Finally, we use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in <cit.>, relying on the formulation of the touching version in terms of the Θ_f operators. 


§ COMBINATORIAL DEFINITIONS


	A square path of size n is a lattice paths going from (0,0) to (n,n) consisting of east or north unit steps, always ending with an east step. The set of such paths is denoted by (n). We call shift of a square path the maximum value s such that  the path intersect the line y=x-s in at least one point. We refer to the line y=x+i-s as i-th diagonal and to the line x=y, (the s-th diagonal) as the main diagonal. A Dyck path is a square path whose shift is 0. The set of Dyck paths is denoted by (n). Of course (n)⊆(n).  


For example, the path in Figure <ref> has shift 3. 




	Let π be a square path of size n. We define its area word to be the sequence of integers a(π) = (a_1(π), a_2(π), ⋯, a_n(π)) such that the i-th vertical step of the path starts from the diagonal y=x+a_i(π). For example the path in Figure <ref> has area word (0,   -3,   -3,   -2,   -2,   -1,   0,   0).



	A partial labelling of a square path π of size n is an element w ∈ℕ^n such that
	
		
  * if a_i(π) > a_i-1(π), then w_i > w_i-1,
		
  * a_1(π) = 0  w_1 > 0,
		
  * there exists an index i such that a_i(π) = - (π) and w_i(π) > 0,
	
	i.e. if we label the i-th vertical step of π with w_i, then the labels appearing in each column of π are strictly increasing from bottom to top, with the additional restrictions that, if the path starts north then the first label cannot be a 0, and that there is at least one positive label lying on the base diagonal.
	
	We omit the word partial if the labelling is composed of strictly positive labels only. 



	A (partially) labelled square path (resp. Dyck path) is a pair (π, w) where π is a square path (resp. Dyck path) and w is a (partial) labelling of π. We denote by (m,n) (resp. (m,n)) the set of labelled square paths (resp. Dyck paths) of size m+n with exactly n positive labels, and thus exactly m labels equal to 0.










The following definitions will be useful later on.


	Let w be a labelling of square path of size n. We define x^w ∏_i=1^n x_w_i|_x_0 = 1.


The fact that we set x_0 = 1 explains the use of the expression partially labelled, as the labels equal to 0 do not contribute to the monomial.

Sometimes we will, with an abuse of notation, write π as a shorthand for a labelled path (π, w). In that case, we use the identification x^π x^w.

Now we want to extend our sets introducing some decorations.


	
	The contractible valleys of a labelled square path π are the indices 1 ≤ i ≤ n such that one of the following holds:
	
		
  * i = 1 and either a_1(π) < -1, or a_1(π) = -1 and w_1 > 0,
		
  * i > 1 and a_i(π) < a_i-1(π),
		
  * i > 1 and a_i(π) < a_i-1(π)  w_i > w_i-1.
	
	
	We define 
    v(π, w) {1 ≤ i ≤ n | i  is a contractible valley},
 corresponding to the set of vertical steps that are directly preceded by a horizontal step and, if we were to remove that horizontal step and move it after the vertical step, we would still get a square path with a valid labelling, with the additional restriction that if the vertical step is in the first row and it is attached to a 0 label, then we require that it is preceded by at least two horizontal steps.



	These slightly contrived conditions on the steps labelled 0 have a more natural formulation in terms of steps labelled ∞, see Section <ref>


This extends the definition of contractible valley given in <cit.> to (partially) labelled square paths.


	
	The rises of a (labelled) square path π are the indices 
    r(π) {2 ≤ i ≤ n | a_i(π) > a_i-1(π)},
 i.e. the vertical steps that are directly preceded by another vertical step. 

	

	A valley-decorated (partially) labelled square path is a triple (π, w, dv) where (π, w) is a (partially) labelled square path and dv ⊆ v(π, w). A rise-decorated (partially) labelled square path is a triple (π, w, dr) where (π, w) is a (partially) labelled square path and dr ⊆ r(π).


Again, we will often write π as a shorthand for the corresponding triple (π, w, dv) or (π, w, dr).

We denote by (m,n)^∙ k (resp. (m,n)^∗ k) the set of partially labelled valley-decorated (resp. rise-decorated) square paths of size m+n with n positive labels and k decorated contractible valleys (resp. decorated rises). We denote by (m,n)^∙ k (resp. (m,n)^∗ k) the corresponding subsets of Dyck paths. 

We also define '(m,n)^∙ k as the set of paths in (m,n)^∙ k such that there exists an index i such that a_i(π) = - (π) and i ∉dv  w_i(π) > 0, i.e. there is at least one positive label lying on the bottom-most diagonal that is not a decorated valley. The importance of this set will be evident later in the paper.

Finally, we sometimes omit writing m or k when they are equal to 0. Notice that, because of the restrictions we have on the labellings and the decorations, the only path with n=0 is the empty path, for which also m=0 and k=0.



We define two statistics on this set that reduce to the ones defined in <cit.> when m=0 and k=0. 


	
	Let (π, w, dr) ∈(m,n)^∗ k and s be its shift. We define 
	
    (π, w, dr) ∑_i ∉dr (a_i(π) + s),

	i.e. the number of whole squares between the path and the base diagonal that are not in rows containing a decorated rise.
	
	For (π, w, dv) ∈(m,n)^∙ k, we define (π, w, dv) (π, w, ∅) ∈(m,n)^∗ 0.


For example, the path in Figure <ref> (left) has area 8. Notice that the area does not depend on the labelling.


	
	Let (π, w, dv) ∈(m,n)^∙ k. For 1 ≤ i < j ≤ n, the pair (i,j) is an inversion if
	
		
  * either a_i(π) = a_j(π) and w_i < w_j (primary inversion),
		
  * or a_i(π) = a_j(π) + 1 and w_i > w_j (secondary inversion),

	
	where w_i denotes the i-th letter of w, i.e. the label of the vertical step in the i-th row. Then we define 
	
    (π) #{ 1 ≤ i < j ≤ n | (i,j)  inversion  j ∉dv } + #{1 ≤ i ≤ n | a_i(π) < 0  w_i > 0 } - # dv

	where again π is a shorthand for (π, w, dv).
	
	For (π, w, dr) ∈(m,n)^∗ k, we define (π, w, dr) (π, w, ∅) ∈(m,n)^∙ 0.


We refer to the middle term, counting the nonzero labels below the main diagonal, as bonus or tertiary dinv.

For example, the path in Figure <ref> (right) has dinv equal to 3: 1 primary inversion in which the leftmost label is not a decorated valley, i.e. (1,7); 1 secondary inversion in which the leftmost label is not a decorated valley, i.e. (1,6); 3 bonus dinv, coming from the rows 3, 4, and 6; 2 decorated valleys. 

It is easy to check that if j ∈ dv then either there exists some inversion (i,j) or a_j < 0. This means that if m=0 the dinv is always non-negative. In fact, thanks to the condition on the first row, the dinv is also non-negative for m>0, and also if π∈'(m,n)^∙ k and it is not a Dyck path, then the dinv is necessarily strictly positive.

Finally, we recall two classical definitions.


	Let p_1, …, p_k be a sequence of integers. We define its descent set 
    𝖣𝖾𝗌(p_1, …, p_k) { 1 ≤ i ≤ k-1 | p_i > p_i+1}
 and its major index (p_1, …, p_k) to be the sum of the elements of the descent set.




§ SYMMETRIC FUNCTIONS
 
For all the undefined notations and the unproven identities, we refer to <cit.>*Section 1, 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 that will appear in our calculations 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 
    _̋μ_̋μ[X] = _̋μ[X; q,t] = ∑_λ⊢ nK_λμ(q,t) s_λ
 the (modified) Macdonald polynomials, where 
    K_λμK_λμ(q,t) = K_λμ(q,1/t) t^n(μ)
 are the (modified) Kostka coefficients (see <cit.>*Chapter 2 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 <cit.>.

If we identify the partition μ with its Ferrers diagram, i.e. with the collection of cells {(i,j)| 1≤ i≤μ_i, 1≤ j≤ℓ(μ)}, then for each cell c∈μ we refer to the arm, leg, co-arm and co-leg (denoted respectively as a_μ(c), l_μ(c), a_μ(c)', l_μ(c)') as the number of cells in μ that are strictly to the right, above, to the left and below c in μ, respectively.

Let M  (1-q)(1-t). For every partition μ, we define the following constants:


    B_μ    B_μ(q,t) = ∑_c ∈μ q^a_μ'(c) t^l_μ'(c), 
    
    	D_μ    D_μ(q,t) = MB_μ(q,t)-1, 
    
    	T_μ    T_μ(q,t) = ∏_c ∈μ q^a_μ'(c) t^l_μ'(c) = q^n(μ') t^n(μ) = e_|μ|[B_μ], 
    Π_μ   Π_μ(q,t) = ∏_c ∈μ / (1,1) (1-q^a_μ'(c) t^l_μ'(c)), 
    
    	w_μ    w_μ(q,t) = ∏_c ∈μ (q^a_μ(c) - t^l_μ(c) + 1) (t^l_μ(c) - q^a_μ(c) + 1).


We will make extensive use of the plethystic notation (cf. <cit.>*Chapter 1).

We need to introduce several linear operators on Λ.



	
	We define the linear operator ∇Λ→Λ on the eigenbasis of Macdonald polynomials as 
    ∇_̋μ = T_μ_̋μ.




	
	We define the linear operator ΠΛ→Λ on the eigenbasis of Macdonald polynomials as 
    Π_̋μ = Π_μ_̋μ
 where we conventionally set Π_∅ 1.



	
	For f ∈Λ, we define the linear operators Δ_f, Δ'_f Λ→Λ on the eigenbasis of Macdonald polynomials as 
    Δ_f _̋μ = f[B_μ] _̋μ,         Δ'_f _̋μ = f[B_μ-1] _̋μ.



Observe that on the vector space of symmetric functions homogeneous of degree n, denoted by Λ^(n), the operator ∇ equals Δ_e_n. 

We also introduce the Theta operators, first defined in <cit.>


	
	For f ∈Λ, we define the linear operator Θ_f Λ→Λ as 
    Θ_f F[X] = Π  f [ X/M] Π^-1 F[X].



It is clear that Θ_f is linear, and moreover, if f is homogenous of degree k, then so is Θ_f, i.e. 
    Θ_f Λ^(n)⊆Λ^(n+k)     for  f ∈Λ^(k).
















It is convenient to introduce the so called q-notation. In general, a q-analogue of an expression is a generalisation involving a parameter q that reduces to the original one for q → 1.


	For a natural number n ∈ℕ, we define its q-analogue as 
    [n]_q 1-q^n/1-q = 1 + q + q^2 + … + q^n-1.



Given this definition, one can define the q-factorial and the q-binomial as follows.


	We define 
    [n]_q! ∏_k=1^n [k]_q   and  nk_q [n]_q!/[k]_q![n-k]_q!




	For x any variable and n ∈∪{∞}, we define the q-Pochhammer symbol as 
    (x;q)_n ∏_k=0^n-1 (1-xq^k) = (1-x) (1-xq) (1-xq^2) ⋯ (1-xq^n-1).



We can now introduce yet another family of symmetric functions.


	
	For 0 ≤ k ≤ n, we define the symmetric function E_n,k by the expansion 
    e_n [ X 1-z/1-q] = ∑_k=0^n (z;q)_k/(q;q)_k E_n,k.



Notice that E_n,0 = δ_n,0. Setting z=q^j we get 
    e_n [ X 1-q^j/1-q] = ∑_k=0^n (q^j;q)_k/(q;q)_k E_n,k = ∑_k=0^n k+j-1k_q E_n,k
 and in particular, for j=1, we get 
    e_n = E_n,0 + E_n,1 + E_n,2 + ⋯ + E_n,n,
 so these symmetric functions split e_n, in some sense.

We care in particular about the following identity.


	
	
    ω(p_n) = ∑_k=1^n [n]_q/[k]_q E_n,k



The Theta operators will be useful to restate the Delta conjectures in a new fashion, thanks to the following results.


	
	
    Θ_e_k∇ e_n-k = Δ'_e_n-k-1 e_n




	
	
    [n]_q/[n-k]_qΘ_e_k∇ω(p_n-k) = [n-k]_t/[n]_tΔ_e_n-kω(p_n)




	
    [n]_t/[n-k]_tΘ_e_k∇ω(p_n-k) = [n-k]_q/[n]_qΔ_e_n-kω(p_n)








§ DELTA CONJECTURES

By Delta conjectures we refer to a family of conjectures that provide a combinatorial interpretation of certain symmetric functions that arise from the Delta operators and show positivity properties. 

The first and most famous of the Delta conjectures is known as shuffle conjecture, now a theorem by E. Carlsson and A. Mellit (see <cit.>).


	
    ∇ e_n = ∑_π∈(n) q^(π) t^(π) x^π









The shuffle theorem is especially important because ∇ e_n has another interpretation, as the bigraded Frobenius characteristic of the S_n module of the diagonal harmonics. This is one of the facts that first motivated the study of Macdonald polynomials, and it has been proved by M. Haiman in <cit.>. See also <cit.> for further details.

The Delta conjecture is a generalisation of the shuffle conjecture, introduced by J. Haglund, J. Remmel, and A. Wilson in <cit.>. In the same paper, the authors suggest that an even more general conjecture should hold, which we call generalised Delta conjecture. It reads as follows.


	
	
    Δ_h_mΔ'_e_n-k-1 e_n = ∑_π∈(m,n)^∙ k q^(π) t^(π) x^π.



For m=0 this conjecture first appears together with the rise version in <cit.>. The full statement, together with a proof of the case q=0, has been given by D. Qiu and A. Wilson in <cit.>.


	
    Δ_h_mΔ'_e_n-k-1 e_n = ∑_π∈(m,n)^∗ k q^(π) t^(π) x^π.



The rise version of the Delta conjecture is simply the case m=0 of the general case.



Recalling that ∇|_Λ^(n) = Δ'_e_n-1|_Λ^(n), it is clear that for k=0 both the versions of the Delta conjecture reduce to the shuffle theorem.









The square conjecture was first suggested by N. Loehr and G. Warrington in <cit.>, and it was then proved by E. Sergel in <cit.> using the shuffle theorem.


	
    ∇ω(p_n) = ∑_π∈(n) q^(π) t^(π) x^π.



Unfortunately, adding zero labels and decorated rises to square paths in the trivial way and q,t-counting the resulting objects with respect to the bistatistic (, ) gives a polynomial that does not match the expected symmetric function. This issue has been addressed by M. D'Adderio and the authors, who stated the generalised Delta square conjecture in <cit.>.


	
    [n-k]_t/[n]_tΔ_h_mΔ_e_n-kω(p_n) = ∑_π∈(m,n)^∗ k q^(π) t^(π) x^π.



where the rise version of the square conjecture is simply the case m=0 of the general case.

The square conjectures used to lack a valley version. Computational evidence suggests the following, checked by computer up to n=6.


	
	
    [n-k]_q/[n]_qΔ_h_mΔ_e_n-kω(p_n) = ∑_π∈(m,n)^∙ k q^(π) t^(π) x^π.



Notice that the symmetric function we propose for the valley version differs from the one appearing in the rise version (for m=0) as the multiplicative factor is the ratio of two q-analogues instead of two t-analogues. This suggests a potential extension of the conjecture to a version with both decorated rises and contractible valleys, possibly using the Theta operators appearing in <cit.>. The power series associated to the obvious combinatorial extension, however, seems to be quasi-symmetric function which is not symmetric, and thus further investigation is required to find suitable statistics.

We can restate it in terms of Theta operators as follows.


	
	
    [n]_t/[n-k]_tΔ_h_mΘ_e_k∇ω(p_n-k) = ∑_π∈(m,n)^∙ k q^(π) t^(π) x^π.



We need to state a refinement of the Delta conjecture, valley version, that naturally arises when stating it in terms of Theta operators. But first, we need another combinatorial definition. Let 
    (m, n \ r)^∙ k{π∈(n)^∙ k|#{ i ∉dv  a_i = - (π)  w_i > 0 } = r },
 which is the set of labelled valley-decorated square paths of size m+n with m labels equal to 0 and k decorations such that there are exactly r steps which are neither 0 labels not decorated valleys on the bottom-most diagonal, and let 
    (m, n \ r)^∙ k(m, n \ r)^∙ k∩(m, n)^∙ k,
	which is the subset of corresponding labelled valley-decorated Dyck paths. We state the following.


	
	
    Θ_e_k∇ E_n-k, r = ∑_π∈(n \ r)^∙ k q^(π) t^(π) x^π.



It is immediate that Conjecture <ref> implies the case m=0 of Conjecture <ref>, as it is enough to sum over r and then apply Theorem <ref>.

We need to state the same refinement for the generalised version too.


	
	
    Δ_h_mΘ_e_k∇ E_n-k, r = ∑_π∈(m, n \ r)^∙ k q^(π) t^(π) x^π.



We now want to state yet another version of the Delta square conjecture, using the set '(m,n)^∙ k previously introduced.


	
	
    Θ_e_k∇ω(p_n-k) = ∑_π∈'(n)^∙ k q^(π) t^(π) x^π.



This conjecture is new and it is more nice-looking than the other forms of the Delta square conjecture as it does not have any multiplicative correcting factor. It also extends nicely to the m>0 case, as follows.


	
	
    Δ_h_mΘ_e_k∇ω(p_n-k) = ∑_π∈'(m,n)^∙ k q^(π) t^(π) x^π.



Our goal is to show that Conjecture <ref> implies Conjecture <ref>, and as a corollary that Conjecture <ref> implies Conjecture <ref>.


§ SCHEDULE NUMBERS FOR REPEATED LABELS




	Let (π, w, dv) be a valley-decorated labelled square path with shift s. For i ≥ 0 we set ρ_i to be the marked word in the alphabet  consisting of the labels appearing in the i-th diagonal, marked with a ∙ if it labels a decorated valley, in increasing order, where we consider c < ∙0 emc <c+1. The diagonal word of (π, w, dv) is (π, w, dv) ρ_ℓ…ρ_1ρ_0.


For example the diagonal word of the path in Figure <ref> is 1243 ∙0 em1 4 1 ∙0 em1. Notice that the ρ_i are the runs of (π, w, dv), i.e. the maximal weakly increasing substrings (disregarding decorations). 





	Let z (π, w, dv) be the diagonal word of a valley-decorated labelled square path (π, w, dv) such that z = ρ_ℓ⋯ρ_0, where the ρ_i's are its runs. We define its i-th run multiplicity functions
	
    z_i    →
    
    		 x    ↦#{ x ∈ρ_i | x   is undecorated}
    
    		z_i^∙   →
    
    		 x    ↦#{ x ∈ρ_i | x   is decorated}

	Notice that each function z_i has finite support. 



	Given z=ρ_ℓ, …, ρ_0 the diagonal word of a square path (π,w) with shift s, where the ρ_i are the runs of z. We set ρ̃_i to be the subword obtained from ρ_i by deleting its decorated numbers and ρ'_i the subword obtained from ρ̃_i by deleting the zeros. Take c ∈ z, for i ∈{0,…, ℓ}, we define its schedule numbers w_i,s(c) as follows: For c ∈
		
    w_i,s(c)    #{d ∈ρ̃_i | d>c } + #{d ∈ρ̃_i-1| d<c }   if   i∈{s+1,…, ℓ}
    #{d ∈ρ̃_i | d>c } + 1 - δ_c,0   if   i=s 
    #{d ∈ρ̃_i | d<c } + #{d ∈ρ̃_i+1| d>c }   if   i∈{0,…, s-1}
    
    			w_i,s^∙ (c)    #{d ∈ρ̃_i | d<c }+#{d ∈ρ̃_i+1| d>c } - δ_c,0δ_i,s-1


Careful! These cardinalities of multisets take into account the multiplicities. 








	
	Let z be a marked word in the alphabet  and let ρ_ℓ, …, ρ_0 be the runs of z, so that z = ρ_ℓ⋯ρ_0. Let b(z,s) ∑_i=0^s-1∑_c ∈ z_i(c) - z_i-1^∙(c) and x^z∏_c∈ zx_c. Then	
	
    ∑_π∈(n)^∙ k
    (π) = s 
    (π) = z  q^(π)t^(π) x^π 
    	= t^(z) q^b(z,s)∏_i=0^ℓ( ∏_c ∈ w_i,s(c) + z_i(c) - 1z_i(c)_q  q^z_i^∙ (c) 2w_i,s^∙ (c)z_i^∙(c)_q )x^z.



The proof of this result is similar to the one described by Haglund and Sergel in <cit.> for Dyck paths, except that we consider repeated labels. For the sake of completeness, we repeat some of their arguments here.


	Let us begin by noting that the right hand side of this equation consists of a finite number of terms different from 1. Indeed z_i(c) = z_i^∙(c) = 0 for all but a finite number of elements of  and thus all but a finite number of q-binomials are equal to 1, which means that the product is actually finite. 
	
	Next, observe that for any π∈(n)^∙ k with (π)= z we trivially have x^π=x^z, so we only need to consider the q,t-enumerators. It is also not difficult to see that for any such path (z)=(π), indeed, 
	
    (π)     = ℓ·#ρ_ℓ + (ℓ-1)·#ρ_ℓ-1 + ⋯ + 1 ·#ρ_1 
        = ρ_ℓ + (ρ_ℓ+ρ_ℓ-1) + ⋯ + (ρ_ℓ+ρ_ℓ-1+⋯ +ρ_1)= (z).
 
	
	For the dinv, we will construct all the paths of a given diagonal word and shift, starting from the empty path, all the while keeping track of the dinv. We do this by applying the procedure described below. For each step, we give illustrate by a (partial) construction of the paths with diagonal word 44223∙0 em3∙0 em3∙0 em011∙0 em2.
	
	First, we construct all the undecorated paths whose diagonal word is ρ̃_s…ρ̃_0. The decorations will be added in the last step.  
	
	
		
  * Starting from the empty path, we insert the numbers of ρ̃_s into the main diagonal, starting with the biggest labels and draw the unique path whose vertical steps are labelled by these numbers. If c is the biggest label in ρ̃_s then w_s,s(c)= 1 and so w_s,s+z_s(c)-1z_s(c)_q=1, indeed there is only one way to insert z_s(c) labels equal to c into the main diagonal and this creates 0 units of dinv because steps with the same label do not create dinv among each other. In general take c a number of ρ̃_s. Then w_s,s(c) is equal to the number of numbers in ρ̃_s bigger than c, that have already be inserted, plus 1 if c ≠ 0. So when inserting z_s(c) numbers equal to c into the diagonal there are w_s,s(c)-1+z_s(c)z_s(c) ways to do it (even if c is 0, as the leftmost label on the main diagonal cannot be a 0, which explains the absence of the extra +1) and the contribution to the dinv is counted by the q-analogue. Indeed we have to choose an interlacing between the w_s,s(c)-1 labels that are already there and the z_s(c) labels equal to c and each time a c precedes a bigger label, one unit of dinv is created.
		
		
		
	
  * Next we add the numbers in ρ̃_s+1, ρ̃_s+2, …, ρ̃_l (in that order) by adding the numbers in ρ̃_i in the i-th diagonal and drawing the unique path with this labelling. As before, we insert the numbers from biggest to smallest in each run. When adding the z_i(c) numbers equal to c of ρ̃_i, we can add them on top of smaller numbers in ρ̃_i-1 or directly northeast of the numbers in ρ̃_i that have already been inserted, i.e. that are bigger. There may also be consecutive c's in the diagonal. So there are w_i,s(c) labels after which the c's may be inserted. So the insertion of the z_i(c) labels equal to c uniquely corresponds to an interlacing of the z_i(c) c's and w_i,s(c) possible insertion positions, that does not start with a c (indeed a c must be inserted after one of the w_i,s(c) positions). Since each time one of the c's precedes one of the w_i,s(c) discussed labels, one unit of dinv is created (primary for bigger labels in ρ̃_i and secondary for smaller labels in ρ̃_i-1), this dinv contribution is q-counted by w_s,i(c)-1+z_i(c)z_i(c)_q.
	
	
	
	
  * Next we add the numbers in runs ρ̃_s-1, ρ̃_s-2,…, ρ̃_0 (in that order), this time from smallest to biggest numbers in each run. As before, we insert the numbers in run ρ̃_i into the i-th diagonal. When adding the z_i(c) numbers equal to c in ρ̃_i, they can either be inserted directly underneath a bigger number from ρ̃_i+1 or directly southwest of a label of ρ̃_i that has already been inserted, i.e. that is smaller. There may also be consecutive c's in the diagonal. So there are again w_s,i(c) places we may insert c. Note that the last c must be underneath a bigger label from ρ̃_i+1, since the path must end with an east step. Thus choosing an interlacing between the w_i,s(c) positions and the z_i(c) c's, ending with a label of the first kind, we get a unique insertion whose dinv is q-counted by w_s,i(c)-1+z_i(c)z_i(c)_qq^δ_c>0∑_i=0^s-1∑_c ∈ z_i(c) since each time an insertion position precedes a c a unit of dinv is created and each positive label underneath the main diagonal creates a unit of bonus dinv.
	
	

	
  * Finally we add the decorated labels. We will insert, for each i, z_i^∙(c) decorated valleys labelled c into the i-th diagonal.   Consider S_i,c, the set of size w_i,s^∙(c), of all the steps that have already been inserted and would create dinv with a the decorated valley labelled c inserted to its right in the i-th diagonal. In other words, the (undecorated) steps in the i+1-th diagonal with a label bigger than c and the (undecorated) steps in the i-th diagonal with a label smaller than c. 
	
	The insertion is slightly different above and below the main diagonal. First, we consider the insertion above the main diagonal, take i∈{s,…, ℓ}.
	Choose a subset T⊆ S_i,c of size z_i^∙(c). We insert one decorated valley labelled c in the i-th diagonal to the right of each element of T and to the left of the next element of S_i,c. We claim that there is always a unique way to do this and that this yields all the possible paths (surjectivity).
		
		Surjectivity and uniqueness. There are two things to show. Firstly, we have to show that a decorated valley cannot be inserted to the left of all the elements is S_i,c. Indeed, we are inserting decorated valleys and decorations must be placed on contractible valleys. This means that if (π, w, dv) is a valley-decorated labelled square path and j∈ dv then
		
			
  * either a_j-1=a_j and w_j-1<w_j, in which case (j-1,j) is a primary inversion.
			
  * or a_j-1> a_j in which case there must be a k<i such that a_k= a_j and a_k+1= a_k +1 (in other words k+1 is a rise). Then either w_k+1> w_j, in which case (k+1,j) is an inversion, or w_k+1≤ w_j, in which case w_k < w_k+1≤ w_j so (k,j) is an inversion. 
		
		
		 It follows that when inserting a decorated valley at least one unit of dinv is created to its left, in other words, at least one element of S_i,c is to its left.  
		
		Secondly, we must argue that there can never be two insertions in between two consecutive elements of S_i,c. In other words, there must always be an element of S_i,c between two decorated valleys labelled c in the i-th diagonal. Indeed 
		
			
  * if one such valley is followed by a vertical step s, its label must be bigger then c and lie in the i+1-th diagonal. Thus, s is an element of S_i,c. 
			
  * if one such valley is followed by a horizontal step the path hits the i-th diagonal at a point p. The next step cannot be another decorated valley or it would not be a contractible valley.  Now we can apply a similar argument as above to the portion of the path starting from p to deduce the existence of an element in S_i,c that lies after p and before the next occurence of the relevant decorated valleys.
		
		This also implies that there exist no two distinct ways of inserting our decorated valley in between two elements of S_i,c. So the insertion must be unique.
		
		Existence. We now show that given an element t of S_i,c, there always exists a way to insert a decorated valley of the discussed kind to its right and to the left of the the next element of S_i,c. It will follow that we can insert the valleys after elements of T one by one and since each must lie strictly between elements of S_i,c these insertions are independent. There some cases and subcases to consider. 
		
		
			
  Case 1.The step t lies in the i-th diagonal, and thus is labelled with a number smaller than c, call it S.
			
				
  Case 1.1 The step t is followed by a vertical step.
					
						
  Case 1.1.1 The label of the vertical step following t, is bigger than c. Call it B. Directly following t, insert a horizontal step followed by a decorated vertical step labelled c. Then continue the path with the vertical step labelled B, since S<c<B, the step labelled c is a contractible valley and the condition on the columns of labelled paths is respected.
						
						
					
						
  Case 1.1.2 The label of the vertical step following t, is smaller than c. Call it S̃. Now consider the portion of the path between the endpoint of t and the first point p where the path crosses the i+1-th diagonal with two consecutive horizontal steps, one ending and one beginning at p (p always exists since the path must return to the main diagonal). If, in this portion of path, there is an occurence of a vertical step contained in the i+1-th diagonal and labelled with a number bigger then c, call it B, then this step is an element of S_i,c. Insert, directly before this step labelled B, a decorated vertical step labelled c and before that step a horizontal step. Since B is bigger than c, the labelling is valid. Furthermore, since the  discussed portion of the path stays weakly above the i+1-th diagonal, the step labelled B must be preceded by a horizontal step and so the inserted valley is preceded by two horizontal steps and thus is contractible.
						 
						 Otherwise, the discussed portion of path does not contain an element of S_i,c. In that case, insert directly after the horizontal step ending at p a horizontal step followed by a decorated vertical step labelled c and continue the path with the horizontal step starting at p. Since the inserted decorated valley is preceded by two horizontal steps, it is always contractible.
						 
						 
					
				
				
  Case 1.2  The step t is followed by a horizontal step. In this case, we may simply insert, directly after t, a horizontal step followed by a decorated vertical step labelled c and continue the path with the horizontal step that followed t. Since S<c, the valley we created is contractible.
				
				
			
		
			
  Case 2. The step t lies in the i+1-th diagonal, and thus is labelled with a number bigger than c, call it B. The reasoning here is very similar to case 1.1.2.: after t the path must cross the i+1-th diagonal. Depending on wether or not there is an element of S_i,c in this portion of path, we have two types of insertions. We include a diagram of each situation.
			
							
		
	
		It follows that the insertion of z_i^∙(c) decorated valleys labelled c into the i-th diagonal is equivalent to the choice of a subset T⊆ S_i,c of size z_i^∙(c). As mentioned above, each such an insertion creates at least one unit of dinv, which can be thought of as being cancelled out by the subtraction of the number of decorated valleys in the computation of the dinv. Additionally, every pair of elements of T creates one unit of dinv, as the decorated valley following the second one creates dinv with the first one. This explains the factor q^z_i^∙(c) 2. Lastly one unit of dinv is created each time an element of S_i,c∖ T precedes an element of T in the path, which explains w_i,s^∙(c)z_i(c)_q
		
		Now for the insertion below the main diagonal, take i∈{0,…, s-1}. The argument is very similar, up to a few minor changes. Indeed, now it is no longer true that any decorated valley must create a unit of dinv with an element to its left. However, there must be an element of S_i,c that is to the right of all the decorated valleys labelled c in the i-th diagonal. Indeed let t be such a decorated valley.
	
	
		
  * if t is followed by a vertical step, it must be contained in the i+1-th diagonal and its label must be bigger then c so it must be an element of S_i,c; 
		
  * if t is followed by a horizontal step and so the path hits the i-th diagonal at a point p, which lies beneath the main diagonal. Since the path must end with an east step above the main diagonal we may deduce the existence of two consecutive vertical steps after p, the first one starting at the i-th diagonal. If the first of these steps has a label smaller that c, it is an element of S_i,c. If it is bigger of equal to c, the succeeding vertical step must have a label bigger than c and thus must be in S_i,c. 
	

	So given a subset T⊆ S_i,c of size z_i^∙(c), we will insert a decorated valley in the i-th diagonal to the left of each element in T and to the right of all the elements of S_i,c preceding the element it. 
	
	 Except for when T contains the very first step of S_i,c, we can recycle the existence of insertion argument given above. Indeed, we can replace all the elements of T with its predecessors in S_i,c, obtaining a set T̃ and apply the described insertions to the right. There is just one subtlety: in that discussion, we used the fact that if the path crosses the i-th diagonal vertically, it must proceed to cross it again, horizontally. This is not necessarily true if i∈{0,…, s-1}. However, as we have shown, there must be at least one element of S_i,c to the right of all the elements of T̃ and this is sufficient to apply the same argument.	 
	 
	 The only thing that is now left to show that, for i < s-1, or i = s-1 and c > 0, it is (uniquely) possible to insert a decorated valley labelled c into the i-th diagonal, to the left of all the elements in S_i,c. If i = s-1 and c=0, then this is not possible. Indeed by definition here cannot be a decorated valley labelled 0 in the first row if the corresponding area letter is -1. Any other decorated valley labelled 0 with area word -1 must be preceded by two horizontal steps (or the valley would not be contractible) forcing a positive label on the main diagonal (which is an element of S_s-1,0) to appear on its left. This restriction explains the presence of the extra -1 in the definition of w_s-1,s(0).
	 
	 If it exists, let us call f the first vertical step of the existing path contained in the i-th diagonal (f is not necessarily an element of S_i,c as its label, call it F, might be bigger than c).
	 
	 
		
  Case 1. The step f is an element of or occurs before every element in S_i,c. Then one of two situations occur. Either f is preceded by a horizontal step, in which case that horizontal step must be preceded by another horizontal step, indeed if is were preceded by a vertical step, this would lie in the i-th diagonal and so f would not be the first. In that case insert a horizontal step followed by a vertical decorated step labelled c in between these two horizontal steps. Or, f is preceded by a vertical step, so the begin point of f is a point were the path crosses the i-th diagonal vertically. Since the path started at (0,0), there must be a point before f where the path crosses this diagonal horizontally, with two consecutive horizontal steps (notice that this point must be unique by definition of f). These two horizontal steps must be preceded by a third horizontal step because if not, this contradicts the definition of f. Insert a horizontal step followed by a decorated vertical step labelled c directly after the first (from the left) of these three horizontal steps.
		
		
		

		
  Case 2. The leftmost element of S_i,c is to the left of all (if any exist) the vertical steps in the i-th diagonal, in which case it must be a step t in the i+1-th diagonal labelled with a number bigger than c, call it B. It follows that t must be preceded by a horizontal step. Insert a horizontal step followed by a decorated vertical step labelled c directly after this horizontal step.
		
				
	 
	 Essentially the same argument as before ensures that there cannot be two decorated valleys inserted between two elements of S_i,c (or before all of its elements). This ensures the unicity of the insertion given a choice of T. Now any two inserted valleys must create one unit of dinv explaining and any element of T succeeding an element of S_i,c∖ T creates a unit of dinv. Each inserted decorated non-zero valley below the diagonal creates a unit of bonus dinv by definition, which we can think of as getting cancelled out with the subtraction of the number of decorated non-zero valleys in the computation of the dinv. Furthermore, for each decorated zero valley below the diagonal, there must be a corresponding non-decorated non-zero valley, which also contributed one unit to the dinv that we can think of as cancelled out with the subtraction of the number of decorated zero valleys. 
	 The contribution to the dinv of this procedure is thus q-counted by q^z_i^∙ (c) 2w_i,s^∙ (c)z_i^∙(c)_q. 
	
	

	
	
	


	





§ THE VALLEY DELTA IMPLIES THE VALLEY SQUARE













Let us define 
    _q,t;x(z,s) ∑_π∈(n)^∙ k
    (π) = s 
    (π) = z q^(π) t^(π) x^π,

where z is the diagonal word of a path in (n)^∙ k. We have the following.


	
	
    _q,t;x(z,s) = q^#ρ'_s-1-z^∙_s-2(0)[#ρ'_s-z_s-1^∙(0)]_q/[#ρ'_s-1z_s-1^∙(0)]_q_q,t;x(z,s-1).




	By Theorem <ref>, we have
	
	
    _q,t;x(z,s) =  t^(z) q^b(z,s)∏_i=0^ℓ( ∏_c ∈ w_i,s(c) + z_i(c) - 1z_i(c)_q  q^z_i^∙ (c) 2w_i,s^∙ (c)z_i^∙(c)_q )x^z.

	
	and
	
	
    _q,t;x(z,s-1) =  t^(z) q^b(z,s-1)∏_i=0^ℓ( ∏_c ∈ w_i,s-1(c) + z_i(c) - 1z_i(c)_q  q^z_i^∙ (c) 2w_i,s-1^∙ (c)z_i^∙(c)_q )x^z.

	
	Now, by definition we have b(z,s) - b(z, s-1) = #ρ'_s-1-z^∙_s-2(0). Also, for c≠ 0 we have that w_i,s^∙(c) does not depend on s and w_i,s^∙(0) = w_i,s-1^∙(0) for i∉{s-2,s-1}. Furthermore for c ∈ we have that w_i,s(c) = w_i,s-1(c) for i ∉{s-1, s}. By simplifying all these terms, we get
	
	
    _q,t;x(z,s) =  q^#ρ'_s-1-z_s-2^∙(0)   ∏_c ∈ w_s,s(c) + z_s(c) - 1z_s(c)_q 
    	 w_s-1,s(c) + z_s-1(c) - 1z_s-1(c)_q 
    	/ w_s,s-1(c) + z_s(c) - 1z_s(c)_q 
    	 w_s-1,s-1(c) + z_s-1(c) - 1z_s-1(c)_q
    		
       ×∏_c∈w_s-1,s^∙(0)z_s-1^∙(0)_q
    	w_s-2,s^∙(0)z_s-2^∙(0)_q
    	/w_s-1,s-1^∙(0)z_s-1^∙(0)_q
    	w_s-2,s-1^∙(0)z_s-2^∙(0)_q
    	_q,t;x(z,s-1)

	
	so all that's left to do is to compute the q-binomials and check that the product yields the desired result.
	
	Recall that
	
    w_s,s(c)     = 1 - δ_c,0 + ∑_a > c z_s(a),            w_s-1,s(c) = ∑_a > c z_s(a) + ∑_a < c z_s-1(a), 
    
    		w_s-1,s-1(c)     = 1 - δ_c,0 + ∑_a > c z_s-1(a)            w_s,s-1(c) = ∑_a > c z_s(a) + ∑_a < c z_s-1(a).
 and that 
	
    w_s-1,s^∙(0)  = #ρ'_s-1        w_s-2,s^∙(0) = ρ'_s-1      
    		w_s-1,s-1^∙(0)  = #ρ'_s        w_s-2,s-1^∙(0) = #ρ'_s-1-1

	
	Let m = max{ c ∈| z_s(c) > 0  or  z_s-1(c) > 0 }. We have
	
    ∏_c ∈ w_s,s(c) + z_s(c) - 1z_s(c)_q/ w_s-1,s-1(c) + z_s-1(c) - 1z_s-1(c)_q  = ∏_c=1^m[z_s-1(c)]_q!/[z_s(c)]_q!·[w_s,s(c) + z_s(c) - 1]_q!/[w_s,s(c) - 1]_q!·[w_s-1,s-1(c) - 1]_q!/[w_s-1,s-1(c) + z_s-1(c) - 1]_q!
        = ∏_c=0^m[z_s-1(c)]_q!/[z_s(c)]_q!·[∑_a ≥ c z_s(a) - δ_c,0]_q!/[∑_a > c z_s(a) - δ_c,0]_q!·[∑_a > c z_s-1(a)-δ_c,0]_q!/[∑_a ≥ c z_s-1(a) - δ_c,0]_q!
        = [∑_a ≥ 0 z_s(a)]_q!/[∑_a > 0 z_s(a)]_q!·[∑_a > 0 z_s-1(a)]_q!/[∑_a ≥ 0 z_s-1(a)]_q!∏_c=0^m[z_s-1(c)]_q!/[z_s(c)]_q!∏_c=1^m[∑_a ≥ c z_s(a)]_q!/[∑_a > c z_s(a)]_q!·[∑_a > c z_s-1(a)]_q!/[∑_a ≥ c z_s-1(a)]_q!
        = [#ρ̃_s-1]_q! [#ρ'_s-1-1]_q!/[#ρ'_s-1]_q![#ρ̃_s-1-1]_q![∑_a ≥ 1 z_s(a)]_q!/[∑_a ≥ 1 z_s-1(a)]_q!·∏_c=0^m[z_s-1(c)]_q!/[z_s(c)]_q!∏_c=1^m[∑_a > c z_s(a)]_q!/[∑_a > c z_s(a)]_q!·[∑_a > c z_s-1(a)]_q!/[∑_a > c z_s-1(a)]_q!
        = [#ρ̃_s-1]_q! [#ρ'_s-1-1]_q!/[#ρ'_s-1]_q![#ρ̃_s-1-1]_q![#ρ'_s]_q!/[#ρ'_s-1]_q!·∏_c=0^m[z_s-1(c)]_q!/[z_s(c)]_q!
        = [#ρ̃_s-1]_q!/[#ρ̃_s-1-1]_q![#ρ'_s]_q/[#ρ'_s-1]_q·∏_c=0^m[z_s-1(c)]_q!/[z_s(c)]_q!

	and
	
    ∏_c ∈ w_s-1,s(c) + z_s-1(c) - 1z_s-1(c)_q/ w_s,s-1(c) + z_s(c) - 1z_s(c)_q    = ∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!·[w_s-1,s(c) + z_s-1(c) - 1]_q!/[w_s-1,s(c) - 1]_q!·[w_s,s-1(c) - 1]_q!/[w_s,s-1(c) + z_s(c) - 1]_q!
        = ∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!·[w_s-1,s(c) + z_s-1(c) - 1]_q!/[w_s,s-1(c) + z_s(c) - 1]_q!
    
    		    = ∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!∏_c=0^m[∑_a > c z_s(a) + ∑_a ≤ c z_s-1(a) - 1]_q!/[∑_a ≥ c z_s(a) + ∑_a < c z_s-1(a) - 1]_q!
        = ∏_c=0^m [∑_a > c z_s(a) + ∑_a ≤ c z_s-1(a) - 1]_q!/∏_c=0^m [∑_a > c-1 z_s(a) + ∑_a ≤ c-1 z_s-1(a) - 1]_q!·∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!
        = ∏_c=0^m [∑_a > c z_s(a) + ∑_a ≤ c z_s-1(a) - 1]_q!/∏_c=-1^m-1 [∑_a > c z_s(a) + ∑_a ≤ c z_s-1(a) - 1]_q!·∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!
        = [∑_a > m z_s(a) + ∑_a ≤ m z_s-1(a) - 1]_q!/[∑_a > -1 z_s(a) + ∑_a ≤ -1 z_s-1(a) - 1]_q!·∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!
        = [∑_a ≤ m z_s-1(a) - 1]_q!/[∑_a ≥ 0 z_s(a) - 1]_q!·∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!
        = [#ρ̃_s-1 - 1]_q!/[#ρ̃'_s - 1]_q!·∏_c=0^m[z_s(c)]_q!/[z_s-1(c)]_q!

	and
	
    w_s-1,s^∙(0)z_s-1^∙(0)_q w_s-2,s^∙(0)z_s-2^∙(0)_q/w_s-1,s-1^∙(0)z_s-1^∙(0)_qw_s-2,s-1^∙(0)z_s-2^∙(0)_q    = [w_s-1,s^∙(0)]_q![w_s-2,s^∙(0)]_q![w_s-1,s-1^∙(0)-z_s-1^∙(0)]_q![w_s-2,s-1^∙(0)-z_s-2^∙(0)]_q!/[w_s-1,s-1^∙(0)]_q![w_s-2,s-1^∙(0)]_q![w_s-1,s^∙(0)-z_s-1^∙(0)]_q![w_s-2,s^∙(0)-z_s-2^∙(0)]_q!
       = [#ρ'_s-1]_q![#ρ'_s-1]_q!/[#ρ'_s]_q![#ρ'_s-1-1]_q![#ρ'_s-z_s-1^∙(0)]_q![#ρ'_s-1-1-z_s-2^∙(0)]_q!/[#ρ'_s-1-z_s-1^∙(0)]_q![#ρ'_s-1-z_s-2^∙(0)]_q!
       =[#ρ'_s-1]_q/[#ρ'_s]_q[#ρ'_s-z_s-1^∙(0)]_q/[#ρ'_s-1-z_s-2^∙(0)]_q.
 
	
	Taking the product we get, after obvious cancellations
	
    [#ρ'_s-z_s-1^∙(0)]_q/[#ρ'_s-1-z_s-2^∙(0)]_q

	which is exactly what we wanted to show.



	If #ρ'_0 ≠ 0, then
	
    _q,t;x(z,s) = q^b(z,s)[#ρ'_s - z_s-1^∙(0)]_q/[#ρ'_0]_q_q,t;x(z).




	It follows immediately by applying <ref> s times and recalling that z_-1^∙(0) = 0.



	
	
    '_q,t;x(n \ r)^∙ k = [n-k]_q/[r]_q_q,t;x(n \ r)^∙ k




	Given a marked word z with ℓ runs and ρ'_0 ≠ 0, we have
	
    ∑_s=0^ℓ_q,t;x(z,s)     = ∑_s=0^ℓ q^b(z,s)[#ρ'_s  - z_s-1^∙(0)]_q/[#ρ'_0]_q_q,t;x(z) 
        = ∑_s=0^ℓ q^b(z,s) [#ρ'_s - z_s-1^∙(0)]_q/[#ρ'_0]_q_q,t;x(z) 
        = [∑_s=0^ℓ (#ρ'_s - z_s-1^∙(0))]_q/[#ρ'_0]_q_q,t;x(z)

	and now taking the sum over all the marked words z of length n with k decorations and ρ'_0 = r, since for any such z, ∑_s=0^ℓ#ρ'_s = n - k + ∑_s=0^ℓ z_s-1^∙(0) (the total number of non-decorated positive labels), the thesis follows immediately.




	If Conjecture <ref> holds, then so does Conjecture <ref>. As a special case, if Conjecture <ref> holds, then so does Conjecture <ref>.



	We recall the statement of Conjecture <ref>, which is
	
    Δ_h_mΘ_e_k∇ E_n-k, r = ∑_π∈(m, n \ r)^∙ k q^(π) t^(π) x^π.

	Applying Corollary <ref>, we have
	
    [n-k]_q/[r]_qΔ_h_mΘ_e_k∇ E_n-k, r = ∑_π∈'(m, n \ r)^∙ k q^(π) t^(π) x^π.

	Summing over r and using Proposition <ref>, we get
	
    Δ_h_mΘ_e_k∇ω(p_n-k) = ∑_π∈'(m,n)^∙ k q^(π) t^(π) x^π,

	as desired.



§ CONCLUDING REMARKS

As we mentioned before, the slightly contrived conditions on the positions of the steps labelled with zeros can be reformulated quite naturally by considering a step labelled 0 as the “pushing” of a step labelled ∞. 


Performing this manoeuvre does not change the dinv (if we define that the ∞'s under the main diagonal do not contribute to the bonus dinv and that there are no ∞'s on the base diagonal). The area changes by a constant factor equal to the number of zeros. 

Several open problems arise from our discussion. There is no interpretation of the symmetric function Δ_h_mΘ_e_k∇ω(p_n-k) in terms of rise-decorated square paths, for which also the schedule formula is lacking. This is one of the very few instances where the valley version seems to be easier to treat than the rise version. Understanding the rise version better might lead to a unified valley-rise conjecture interpreting Θ_e_jΘ_e_k∇ e_n-k-j. 

Lastly, it would be nice to show that the valley Delta conjecture implies the generalised valley Delta conjecture. Given that, our results would be conditional only on the valley Delta conjecture. There might be a way to prove this using the “pushing” manoeuvre described above to interpret the behaviour of the h_j^⊥ operator. We have some symmetric function identities suggesting that this avenue might be fruitful, and some of these conjectural identities are strongly suggested by certain relations among the combinatorial objects.





amsalpha


