Strong factorization property of Macdonald polynomials and higher-order Macdonald's positivity conjecture

We prove a strong factorization property of interpolation Macdonald polynomials when $q$ tends to $1$. As a consequence, we show that Macdonald polynomials have a strong factorization property when $q$ tends to $1$, which was posed as an open question in our previous paper with F\'eray. Furthermore, we introduce multivariate $q,t$-Kostka numbers and we show that they are polynomials in $q,t$ with integer coefficients by using the strong factorization property of Macdonald polynomials. We conjecture that multivariate $q,t$-Kostka numbers are in fact polynomials in $q,t$ with nonnegative integer coefficients, which generalizes the celebrated Macdonald's positivity conjecture.

introduced a new family of symmetric functions J (q,t) λ (x) depending upon a partition λ, a set of variables x = {x 1 , . . . , x N } and two real parameters q, t. They were immediately hailed as a breakthrough in symmetric function theory as well as special functions, as they contained most of the previously studied families of symmetric functions such as Schur polynomials, Jack polynomials, Hall-Littlewood polynomials and Askey-Wilson polynomials as special cases. They also satisfied many exciting properties, among which we just mention one, which led to a remarkable relation between Macdonald polynomials, representation theory, and algebraic geometry. This property, called Macdonald's postivity conjecture [Mac88], states that the coefficients K (q,t) µ,λ in the expansion of J (q,t) λ (x) into the "plethystic Schur" basis s µ [X(1−t)] (for the readers not familiar with the plethystic notation we refer to [Mac95,Chapter VI.8]) are polynomials in q, t with nonnegative integer coefficients. Garsia and Haiman [GH93] refined this conjecture, giving a representation theoretic interpretation for the coefficients in terms of Garsia-Haiman modules, an interpretation which was finally proved almost ten years later by Haiman [Hai01], who connected the problem to the study of the Hilbert scheme of N points in the plane from algebraic geometry. It quickly turned out that Macdonald polynomials have found applications in special function theory, representation theory, algebraic geometry, group theory, statistics, quantum mechanics, and much more [GR05].
Moreover, their fascinating and rich combinatorial structure is one of the most important object of interest in contemporary algebraic combinatorics.

Strong factorization property of interpolation Macdonald polynomials.
The main goal of this paper is to state and partially prove a generalization of celebrated Macdonald's positivity conjecture. We are going to do by proving that Macdonald polynomials have strong factorization property when q → 1, which also resolves the problem posed by the author of this paper and Féray in our recent joint paper [DF16,Conjecture 1.5].
Instead of proving Theorem 1.3, we prove a stronger result that interpolation Macdonald polynomials have a small cumulant property when q → 1, from which Theorem 1.3 follows as a special case. To make this section complete, let us introduce interpolation Macdonald polynomials.
Interpolation polynomials are characterized by certain vanishing condition. Sahi [Sah96] proved that for each partition λ of length ℓ(λ) ≤ N , there exists a unique (inhomogenous) symmetric polynomial J (q,t) λ (x) of degree |λ|, where x = (x 1 , . . . , x N ), which has following properties: This symmetric polynomial is called interpolation Macdonald polynomial and it has a remarkable property which explains its name: its top-degree part is equal to Our main result is the following theorem: Theorem 1.4. Let λ 1 , · · · , λ r be partitions. Then we have a following small cumulant property when q → 1: where κ J (λ 1 , · · · , λ r ) is a cumulant of interpolation Macdonald polynomials.
1.3. Generalized Macdonald's positivity conjecture. As we already mentioned, the purpose of this paper is to generalize q, t-Kostka numbers and to prove that they are polynomials in q, t with integer coefficients. Before we define generalized q, t-Kostka numbers we just mention that strictly from the definition of q, t-Kostka numbers, they are elements of Q(q, t), and it took six or seven years after Macdonald formulated his conjecture to prove that they are in fact polynomials in q, t with integer coefficients, which was proved independently by many authors [GR98,GT96,KN98,Kno97,LV97,Sah96]. This result will be important to prove integrality of the generalized q, t-Kostka numbers.
Let λ 1 , · · · , λ r be partitions. We define generalized q, t-Kostka numbers K (q,t) µ;λ 1 ,...,λ r by the following equation Note that when r = 1, the generalized q, t-Kostka number K (q,t) µ;λ 1 is equal to the ordinary q, t-Kostka number K (q,t) µ,λ with λ 1 = λ. In particular, integrality of Littlewood-Richardson coefficients together with the integrality result on q, t-Kostka numbers implies that Thus applying Theorem 1.3 into the above result, we obtain immediately the following theorem: Theorem 1.5. Let λ 1 , . . . , λ r be partitions. Then, for any partition µ, the generalized q, t-Kostka number K (q,t) µ;λ 1 ,...,λ r is a polynomial in q, t with integer coefficients. We recall that Macdonald's positivity conjecture is a well-established theorem nowadays since Haiman proved positivity in 2001 [Hai01]. We ran some computer simulations which suggested that generalized q, t-Kostka numbers are also polynomials with positive coefficients. Unfortunately, we are not able to prove it, since our techniques of the proof of Theorem 1.4 does not seem to be applicable to this problem and we state it in this paper as a conjecture. Conjecture 1.6. Let λ 1 , . . . , λ r be partitions. Then, for any partition µ, the generalized q, t-Kostka number K (q,t) µ;λ 1 ,...,λ r is a polynomial in q, t with positive, integer coefficients.
1.4. Related problems. We finish this section, mentioning some similar or somewhat related problems. First, we recall that one of the most typical application of cumulant is to show that certain family of random variables is asymptotically Gaussian. Especially, when one deals with discrete structures, the main technique is to show that cumulants have certain small cumulant property, which is in the same spirit as our Theorem 1.3; see [Śni06,FM12,Fér13,DŚ16]. It is therefore natural to ask for a probabilistic interpretation of Theorem 1.3. In particular, does it lead to some kind of central limit theorem? The most natural framework to investigate this problem seems to be related with Macdonald processes introduced by Borodin and Corwin [BC14] or representation-theoretical interpretation of Macdonald polynomials given by Haiman [Hai01].
A second problem is related to the combinatorics of Jack polynomials, which are special cases of Macdonald polynomials. In fact, Theorem 1.3 was posed as an open question in our previous paper joint with Féray [DF16], where we proved that Jack polynomials have strong factorization property when α → 0, where α is a Jack-deformation parameter. In the same paper we use this result as a key tool to prove polynomiality part of the so-called b-conjecture, stated by Goulden and Jackson [GJ96]. This conjecture says that certain multivariate generating function involving Jack symmetric functions expressed in power-sum basis gives rise to multivariate generating function of bipartite maps (bipartite graphs embedded into some surface), where exponent of β := α − 1 has an interpretation as some mysterious "measure of non-orientability" of the associated map. The conjecture is still open, while some special cases have been solved [GJ96, La 09, KV16,Doł16]. It is very tempting to build a q, t-framework which will generalize b-conjecture. Although we can simply replace Jack polynomials by Macdonald polynomials in the definition of multivariate generating function given by Goulden and Jackson and use the same techniques as in [DF16] to prove that expanding it in properly normalized power-sum basis we obtain polynomials in q, t, we do not obtain positive, neither integer coefficients. Therefore, we leave wide-open a question of the possibility of building a proper framework which generalizes b-conjecture to two parameters in a way that it is related to counting some combinatorial objects.
1.5. Organization of the paper. We describe all necessary definitions and background in Section 2. Section 3 gives the proof of Theorem 1.4 which is preceded by an explanation of the main idea of the proof. In Section 4 we discuss cumulants and their relation with strong factorization property, and we investigate a relation between cumulants and derivatives that is in the heart of the proof of Theorem 1.4. Finally, Section 5 is devoted to the proof of two intermediate steps of the proof of Theorem 1.4.

PRELIMINARIES
2.1. Set partitions lattice. The combinatorics of set partitions is central in the theory of cumulants and will be important in this article. We recall here some well-known facts about them.
A set partition of a set S is a (non-ordered) family of non-empty disjoint subsets of S (called parts of the partition), whose union is S. In the following, we always assume that S is finite.
Denote P(S) the set of set partitions of a given set S. Then P(S) may be endowed with a natural partial order: the refinement order. We say that π is finer than π ′ (or π ′ coarser than π) if every part of π is included in a part of π ′ . We denote this by π ≤ π ′ .
Endowed with this order, P(S) is a complete lattice, which means that each family F of set partitions admits a join (the finest set partition which is coarser than all set partitions in F ; we denote the join operator by ∨) and a meet (the coarsest set partition which is finer than all set partitions in F ; we denote the meet operator by ∧). In particular, the lattice P(S) has a maximum {S} (the partition in only one part) and a minimum {{x}, x ∈ S} (the partition in singletons).
Lastly, denote µ the Möbius function of the partition lattice P(S). Then, for any pair π ≤ σ of set partitions, the value of the Möbius function has a product form: where the product is taken over all blocks of a partition σ, and for a given block denotes a Möbius function of the lattice P(B ′ ) of the interval in between a partition {B ∈ π : B ⊂ B ′ }, and a maximal element {B ′ }. This function is given by an explicit formula where #π denotes the number of parts of π. We finish this section by stating a well-known result on computing a Möbius functions of lattices.
2.2. Partitions. We call λ := (λ 1 , λ 2 , . . . , λ l ) a partition of n if it is a weakly decreasing sequence of positive integers such that λ 1 + λ 2 + · · · + λ l = n. Then n is called the size of λ while l is its length. As usual we use the notation λ ⊢ n, or |λ| = n, and ℓ(λ) = l. We denote the set of partitions of n by Y n and we define a partial order on Y n , called dominance order, in the following way: Then, we extend the notion of dominance order on the set of partitions of arbitrary size by saying that λ µ ⇐⇒ |λ| < |µ|, or |λ| = |µ| and λ ≤ µ.
For any two partitions λ ∈ Y n and µ ∈ Y m we can construct a new partition λ ⊕ µ ∈ Y n+m by setting λ ⊕ µ := (λ 1 + µ 1 , λ 2 + µ 2 , . . . ). Moreover, there exists a canonical involution on the set Y n , which associate with a partition λ its conjugate partition λ t . By definition, the j-th part λ t j of the conjugate partition is the number of positive integers i such that λ i ≥ j. A partition λ is identified with some geometric object, called Young diagram, that can be defined as follows: Finally, we define two combinatorial quantities associated with partitions that we will use extensively through this paper. First, we define (q, t)-hook polynomial h (q,t) (λ) by the following equation We also introduce a partition binomial given by

Interpolation Macdonald polynomials as eigenfunctions.
We already defined interpolation Macdonald polynomials in Section 1.2, but we are going to introduce another, equivalent definition that is more convenient in the framework of the following paper. Since this is now a well-established theory, results of this section are given without proofs but with explicit references to the literature (mostly to Macdonald's book [Mac95] and Sahi's paper [Sah96]). First, consider the vector space Sym N of symmetric polynomials in N variables over Q(q, t). Let T q,x i be the "q-shift operator" defined by Let us define an operator Proposition 2.2. There exists a unique family J (q,t) λ (indexed by partitions λ of length at most N ) in Sym N that satisfies: These polynomials are called interpolation Macdonald polynomials. This is a result of Sahi [Sah96]. His original definition requires that the coefficients a λ ν are only rational functions in q, t with rational coefficients, but in the same paper Sahi proved that they are in fact polynomials in q, t −1 , t (and even in q, t when |ν| = |λ|) with integer coefficients, which will be important for us later. We just add for completness of the presentation that we are using different notation and normalization than Sahi, so function (x) with our notation, and c λ (q, t) from Sahi's paper is the same as h (q,t) (λ) with our notation.
Above definition says that the interpolation Macdonald polynomial J (q,t) λ depends on the parameter N , that is the number of variables. However, one can show that it satisfies the compatibility relation J and thus J (q,t) λ can be seen as a symmetric function. In the sequel, when working with differential operators, we sometimes confuse a symmetric function f with its restriction f (x 1 , . . . , x N , 0, 0, . . . ) to N variables.
It was shown by Macdonald [Mac95, Chapter VI, (3.9)-(3.10)] that  Plugging it into Eq. (5) we observe that: Note that we can expand an operator D around q = 1 as a linear combination of differential operators in the following form: . As a consequence we have a following identity: where ∂ q is a partial derivative with respect to q, b N j (λ) is given by Eq. (4), and Corollary 2.3. Let f ∈ Sym be a symmetric function with an expansion in the monomial basis of the following form: where λ is a fixed partition, and d µ ∈ Q(t). If, for any number N of variables,

POLYNOMIALS
In this section we prove Theorem 1.4. Since its proof involves many intermediate results which can be considered as independent of Theorem 1.4, we believe that presenting them before the proof of the main result might discourage the reader, and we decided to explain the main idea of the proof of Theorem 1.4 first, then give the proof with all the details, and finally present all the remaining proofs of the intermediate results in the separate sections.
Proof of Theorem 1.4. We recall that we need to prove that for any positive integer r, and for any partitions λ 1 , . . . , λ r we have a following bound for the cumulant: The proof will by given by induction on r. The fact that Macdonald interpolation polynomials J (q,t) λ have no singularity in q = 1 is straightforward from the result of Sahi presented in Proposition 2.2. That covers a case r = 1. Now, notice that for any ring R, and any rational function f ∈ R[q], the following conditions are equivalent Thus, we are going to prove that where κ J (λ 1 , . . . , λ r ) denotes the cumulant with parameters q −1 , t −1 . From now on, until the end of this proof, κ J (λ 1 , . . . , λ r ) denotes the cumulant with parameters q −1 , t −1 .
Let R be a ring, and let f ∈ R[q, q −1 ] be a Laurent polynomial in q. We introduce a following notation: for any nonnegative integer k the coefficient [(q − 1) k ]f ∈ R is defined by the following expansion: where deg(f ) is the smallest possible nonnegative integer such that It is clear that for two Laurent polynomials f, g ∈ R[q, q −1 ] and nonnegative integer k one has the following identity: With the above notation, we have to prove that for any integer 0 ≤ k ≤ r − 2 the following equality holds true: Notice now that expansion of f into monomial basis involves only monomials m µ indexed by partitions µ ≺ λ [r] , which is ensured by Proposition 4.8. Thus, if we are able to show that the following equation holds true: )f, then f = 0 by Corollary 2.3, and the proof is over. So our goal is to prove Eq. (8).
In order to do that we make a following observation: an interpolation Macdonald polynomial J (q −1 ,t −1 ) is an eigenfunction of the operator D. Since the cumulant is a linear combination of products of interpolation Macdonald polynomials it will be very convenient if the action of D on such product will be given by Leibniz rule. Unfortunately, it is not the case. However, the trick is to decompose Dκ J (λ 1 , . . . , λ r ) into two parts: the first part is given by "forcing" Leibniz rule for the action of D on the product of interpolation Macdonald polynomials, and the second part is given by the difference between the proper action of D on cumulant, and between the forced version. To be more precise and This decomposition turned out to be crucial. Indeed, Lemma 5.2 ensures that the first part can be expressed as a linear combination of products of cumulants of less then r element, thus we can use an induction hypothesis to analyze it. Similarly, Lemma 5.3 states that the second part can be given by an expression involving products of cumulants of less then r elements, and again, an inductive hypothesis can be used to its analysis. Then, comparing the coefficient of (q − 1) k in the left hand side of Eq. (9) with the coefficient of (q − 1) k in the right hand side of Eq. (9) we obtain Eq. (8). Let us go into details. Expanding operator D around q = 1 (see Eq. (6)) we have that Moreover, applying Lemma 5.2, we have that Finally, applying Lemma 5.3, we obtain a following identity We recall that the right hand side (RHS for short) of Eq. (10) is equal to the sum of right hand sides of Eq. (11) and Eq. (12). Let k = 1. Then RHS of Eq. (10) is equal to

CUMULANTS
In this section we introduce cumulants and we investigate an action of derivations on them, which is crucial in the proof of ??. We also explain the connection between strong factorization property and small cumulant property, and we present some applications of it relevant for our work. We begin with some definitions.

Partial cumulants.
Definition 4.1. Let (u I ) I⊆J be a family of elements in a field, indexed by subsets of a finite set J. Then its partial cumulant is defined as follows. For any non-empty subset H of J, set where µ is a Möbius function of the set-partition lattice; see Section 2.1. The terminology comes from probability theory. Let J = [r], and let X 1 , . . . , X r be random variables with finite moments defined on the same probability space. Then define u I = E( i∈I X i ), where E denotes the expected value. The quantity κ [r] (u) as defined above, is known as the joint (or mixed) cumulant of the random variables X 1 , . . . , X r . Also, κ H (u) is the joint/mixed cumulant of the smaller family {X h , h ∈ H}.
Joint/mixed cumulants have been studied by Leonov and Shiryaev in [LS59] (see also an older note of Schützenberger [Sch47], where they are introduced under the French name déviation d'indépendence). They now appear in random graph theory [JŁR00, Chapter 6] and have inspired a lot of work in noncommutative probability theory [NŚ11].
A classical result -see, e.g., [JŁR00, Proposition 6.16 (vi)] -is that relation Eq. (13) can be inverted as follows: for any non-empty subset H of J,

Derivations and cumulants.
Let R be a ring. We define an R-module of derivations Der K which consists of linear maps D : R → R satisfying a following Leibniz rule: For any positive integers r, k, and for any elements f 1 , . . . , f r ∈ R we define Let K be a field, and D ∈ Der K be a derivation. Then, for any family u = (u I ) I⊆[r] of elements in a field K we define the following deformed action of D k on the cumulant: The following lemma will be crucial for proving our main result.

Lemma 4.2.
For any positive integers r, k, for any family u = (u I ) I⊆[r] of elements in a field K and for any derivation D ∈ Der K , the following identity holds true: Here, N π + denotes the set of functions α : π → N + , the symbol |α| is defined as Proof. First of all, notice that for any elements f 1 , . . . , f r ∈ K, and for any positive integer k the following generalized Leibniz rule holds true: which is easy to prove by induction (D 0 := Id by convention). Notice now that the both hands of Eq. (15) are linear combinations of elements of the form where π ∈ P([r]), and α ∈ N π is a composition of k. Let us call RHS the righthand side of Eq. (15), and analogously LHS the left-hand side of Eq. (15). Let us fix a set-partition π ∈ P([r]), and a composition α ∈ N π of k. We would like to show that We define a support supp(α) of α in a standard way: We now analyze the coefficient We can see that a nonzero contribution comes from the elements of the following form: σ ≥ π, and for each element B ′ ∈ σ there exists an element B ∈ supp(α) such that B ′ ⊂ B ′ . In other terms, σ is a partition which has a property that σ ≥ π, and where partition τ is constructed from π by merging all its blocks lying in a support of α, i.e. : (18) τ := supp(α) ∪ (π \ supp(α)) .
Using definition of cumulant Eq. (13), and formula Eq. (16), we can compute the coefficient Plugging it into Eq. (15), we obtain that where τ is a partition given by formula Eq. (18). Here, the last equality is a consequence of the formula Eq. (2) for the Möbius function µ(π, σ). Now, notice that a partition τ is constructed in a way that τ ≥ π, and inequality is strict whenever # supp(α) > 1. Thus, we can apply Proposition 2.1 to get otherwise.
Comparing it with Eq. (17), we can see that which finishes the proof.

A multiplicative criterion for small cumulants.
Let R be a ring and q a formal parameter. We consider a family u = (u I ) I⊆[r] of elements of R(q) indexed by subsets of [r]. Throughout this section, we also assume that these elements are non-zero and u ∅ = 1.
In addition to partial cumulants, we also define the cumulative factorization error terms T H (u) of the family u. The quantities T H (u) H⊆[r],|H|≥2 are inductively defined as follows: for any subset G of [r] of size at least 2, (1 + T H (u)).
Using inclusion-exclusion principle, a direct equivalent definition is the following: for any subset H of [r] of size at least 2, set Féray (using different framework) [Fér13] proved a following statement, which was reproved in our recent joint paper with Féray [DF16, Proposition 2.3] using the framework of the current paper: Remark. In fact, above proposition was proved in the case r = 0, but it is enough to shift indeterminate q → q − r to obtain a general result.
A first consequence of this multiplicative criterion for small cumulants is the following stability result. Proof. This is trivial for the strong factorization property and the small cumulant property is equivalent to it.
Here is another consequence:  . Let c ∈ N and (c i ) i∈K be a family of some nonnegative integers, and let C = 1 ∈ R. For a subset I of K, we define v I = 1 − C · q c+ i∈I c i Then we have, for any subset H of K, Proof. It is enough to prove the statement for H = K. Indeed, the case of a general set H follows by considering the same family restricted to subsets of H.
Define R ev (resp. R odd ) as where the product runs over subsets of K of even (resp. odd) size. WLOG, we can assume that |K| is even (the case when |K| is odd is analogous). With this notation, T K (v) = R ev /R odd − 1 = (R ev − R odd )/R odd . Since R −1 odd = O 1 (1) (each term in the product is O 1 (1), as well as its inverse), it is enough to show that It is clear that Let us fix a positive integer l < |K|. Expanding the product in the definition of R ev in the basis {(q − 1) j } j≥0 , and using binomial formula, one gets The index set of the second summation symbol is the list of sets of i distinct (but not necessarily disjoint) subsets of K of even size, and The factor 1 i! in the above formula comes from the fact that we should sum over sets of i distinct subsets of K, instead of lists, but it is the same as the summation over the set of lists of i distinct subsets of K and dividing by the number of permutations of [i]. Strictly from this formula it is clear that [(q − 1) l ]R ev is a symmetric polynomial in c i : i ∈ K of degree at most l. Of course, a similar formula with subsets of odd size holds for [(q − 1) l ]R odd , which shows that it is a symmetric polynomial in c i : i ∈ K of degree at most l, as well. For any positive integers n, k we define a set Y(n, k) of sequences of n nonnegative, nonincreasing integers, which are of the following form: It is well known (see for example [KS96,Theorem 2.1]) that if f, g are two symmetric polynomials of degree at most k in n indeterminates, then Thus, in order to show that [(q − 1) l ]R ev = [(q − 1) l ]R odd it is enough to show that this equality holds for all (c i ) i∈K ∈ Y(|K|, l). Note that since l < |K|, then c k is necessarily equal to 0, where k is the biggest possible k ∈ K. It means that a function f : (K) ev := {δ ⊂ K : δ has even size } → (K) odd := {δ ⊂ K : δ has odd size } given by f (δ) := δ∇{k}, where ∇ is the symmetric difference operator, is a bijection which preserves the following statistic |δ| c = |f (δ)| c .
Thus one has Since l < |K| was an arbitrary positive integer, we have shown that which finishes the proof. Proof. Fix some subset I = {i 1 , . . . , i t } of [r] with i 1 < · · · < i t . Observe that the Young diagram λ I can be constructed by sorting the columns of the diagrams λ i 1 , . . . , λ it in decreasing order. When several columns have the same length, we put first the columns of λ i 1 , then those of λ i 2 and so on; see Fig. 2 (at the moment, please disregard symbols in boxes). This gives a way to identify boxes of λ I with boxes of the diagrams λ is (1 ≤ s ≤ t) that we shall use below. With this identification, if b = (c, r) is a box in λ g for some g ∈ I, its leg-length in λ I is the same as in λ g . We denote it by ℓ(b).
However, the arm-length of b in λ I may be bigger than the one in λ g . We denote these two quantities by a I (b) and a g (b). Let us also define a i (b) for i = g in I, as follows: • for i < g, a i (b) is the number of boxes b ′ in the r-th row of λ i such that the size of the column of b ′ is smaller than the size of the column of b (e.g., on Fig. 2, for i = 1, these are boxes with a diamond); • for i > g, a i (b) is the number of boxes b ′ in the r-th row of λ i such that the size of the column of b ′ is at most the size of the column of b (e.g., on Fig. 2, for i = 3, these are boxes with an asterisk). Looking at Fig. 2, it is easy to see that Therefore, for G ⊆ [r], one has: From the definition of T [r] (u), given by Eq. (19), we get: The expression inside the bracket corresponds to 1 + T Plugging Eq. (20) into definition of v b I , we observe that v b I is as in Lemma 4.6 with the following values of the parameters: K = [r] \ {g}, C = t ℓ(b)+1 , c = 1, and c i = a i (b) for i = g. Therefore, we conclude that Going back to Eq. (21), we have: which completes the proof.
We finish this section by presenting an important corollary from the above result.

DIFFERENTIAL OPERATOR AND CUMULANT OF INTERPOLATION MACDONALD POLYNOMIALS
Let us fix partitions λ 1 , . . . , λ r , and for any subset I ⊆ [r] we define u I := J (q −1 ,t −1 ) λ I . The purpose of this section is an analysis of the action of the differential operator D -defined in Eq. (5) -on the cumulant κ J (λ 1 , . . . , λ r ) = κ [r] (u) with parameters q −1 , and t −1 . In particular, this analysis leads to the proofs of two crucial lemmas used in the proof of Theorem 1.4. 5.1. Analysis of the decomposition. For any positive integer r and for any partitions λ 1 , . . . , λ r we define where b N j λ I is given by Eq. (4). Proposition 5.1. Let r > j ≥ 1 be positive integers. Then, for any partitions λ 1 , . . . , λ r one has: InEx j (λ 1 , . . . , λ r ) = 0.
Proof. Expanding the definition and completing partitions with zeros, we have: In particular, we have to prove that the summand corresponding to any given 1 ≤ i ≤ N is equal to 0. In other terms, we have to show that a polynomial

I⊆[r]
(−1) r−|I| x I j = 0, where x = (x 1 , . . . , x r ), and x I := i∈I x i . Note that it is a symmetric polynomial in x without constant term of degree at most j, thus it is enough to show that the coefficient of x µ := x µ 1 1 · · · x µr r is equal to zero for all nonempty partitions µ of size at most j. This coefficient is given by: (x) j := x(x − 1) · · · (x − j + 1) = 0≤k≤j s(j, k)x k .
Since ℓ(µ) ≤ |µ| ≤ j < r, we have that where D J Proof. Note that strictly from the definition of interpolation Macdonald polynomials given by Proposition 2.2 we know that for any pair partition π ∈ P([r]) the following identity holds: If we substitute it into definition of Dκ J (λ 1 , . . . , λ r ), we have that Fix a set partition σ ∈ P([r]). We claim that the expression in the bracket in the above equation is given by the following formula where S(n, k) is the Stirling number of the second kind and the last equality comes from the relation 0≤k≤n S(n, k)(x) k = x n evaluated at x = −1 (here, (x) k := x(x − 1) · · · (x − k + 1) denotes the falling factorial). This finishes the proof of Eq. (22), and also completes the proof of the lemma.