Algebras with two multiplications and their cumulants

Cumulants are a notion that comes from the classical probability theory, they are an alternative to a notion of moments. We adapt the probabilistic concept of cumulants to the setup of a linear space equipped with two multiplication structures. We present an algebraic formula which involves those two multiplications as a sum of products of cumulants. In our approach, beside cumulants, we make use of standard combinatorial tools as forests and their colourings. We also show that the resulting statement can be understood as an analogue of Leonov--Shiraev's formula. This purely combinatorial presentation leads to some conclusions about structure constant of Jack characters.


Cumulants in probability theory.
One of classical problems in probability theory is to describe the joint distribution of a family (X i ) of random variables in the most convenient way. Common solution of this problem is to use the family of moments, i.e. the expected values of products of the form E (X i1 · · · X i l ) .
It has been observed that in many problems it is more convenient to make use of the cumulants [Hal81,Fis28], defined as the coefficients of the expansion of the logarithm of the multidimensional Laplace transform around zero: (1) κ (X 1 , . . . , X n ) := [t 1 · · · t n ] log Ee t1X1+···+tnXn = ∂ n ∂t 1 · · · ∂t n log Ee t1X1+···+tnXn where the terms on the right-hand side should be understood as a formal power series in the variables t 1 , . . . , t n . Cumulant is a linear map with respect to each of its arguments. There are some good reasons for claiming advantage of cumulants over the moments. One of them is that the convolution of measures corresponds to the product of the Laplace transforms or, in other words, to the sum of the logarithms of the Laplace transforms. It follows that the cumulants behave in a very simple way with respect to the convolution, namely cumulants linearize the convolution.
Cumulants allow also a combinatorial description. One can show that the expression (1) is equivalent to the following system of equations, called the momentcumulant formula: which should hold for any choice of the random variables X 1 , . . . , X n whose moments are all finite. The above sum runs over the set partitions ν of the set [n] = {1, . . . , n} and the product runs over the blocks of the partition ν.
Observe that the moment-cumulant formula defines the cumulant κ( X 1 , . . . , X n ) inductively according to the number of arguments n.

∈ B
where the terms on the right-hand side should be understood as in Eq. (1). In this general approach, cumulants give a way of measuring the discrepancy between the algebraic structures of A and B.

Framework.
In the this paper we are interested in a following particular case. We assume that A is a linear space equipped with two commutative multiplication structures, which correspond to two products: · and * . Together with each multiplication A form the commutative algebra. We call such structure an algebra with two multiplications. We also assume that the mapping E is the identity map on A: In this case the cumulants measure the discrepancy between these two multiplication structures on A. This situation arises naturally in many branches of algebraic combinatorics, for example in the case of Macdonald cumulants [Doł17a,Doł17b] and cumulants of Jack characters [DF17,Śn16].
Since the mapping E is the identity, we can define cumulants of cumulants and further compositions of them. The terminology of cumulants of cumulants was introduced in [Spe83] and further developed in [Leh13] (called there nested cumulants) in a slightly different situation of an inclusion of algebras C ⊆ B ⊆ A and conditional expectations A E1 −→ B E2 −→ C.
As we already mentioned in Section 1.1, cumulants allow also a combinatorial description via the moment-cumulant formula. When E is the identity map (3) is equivalent to the following system of equations: (4) a 1 * · · · * a n = ν b∈ν for any a i ∈ A (the product on the right-hand side is the ·-product). The above sum runs over the set partitions ν of the set [n] and the product runs over the blocks of the partition ν.
Let A be a multiset consisting of elements of the algebra A. To simplify notation, for any partition ν of a multiset A we introduce the corresponding cumulant κ ν as the product: We denote by P (A) the set of all partitions of A. With this notation, the momentcumulant formula has the following form: Example 1.2. Given three elements a 1 , a 2 , a 3 ∈ A, we have: a 2 , a 3 ).

The main result.
The purpose of this paper is to present an algebraic formula which involves two multiplications on linear space A: a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn , as a sum of products of only one type of multiplication. We use the following notation. We denote by A 1 , . . . , A n multisets consisting of elements of A. We denote by A = A 1 ∪ · · · ∪ A n the multiset, corresponding to the sum of all multisets A i . We use also the following notation for elements of A i : . . , a i ki , hence the multiset A consists of the following elements: 1 , . . . , a 1 k1 , . . . , a n 1 , . . . , a n kn . Due to a combinatorial nature of this result we introduce now the definitions of the mixing reduced forests and theirs cumulants. We begin with the following definition.
Definition 1.3. Consider a forest F whose leaves are labelled by elements of an algebra A. We denote by A the multiset consisting of labels of all leaves. If each node (vertex which is not a leaf) of F , has at least two descendants, we call F a reduced forest with leaves in A. We denote the set of such forests by F (A) (see Figure 1).
For a reduced forest F ∈ F (A) we associate a cumulant κ F in the following way: Six of them (on the right-hand side) are mixing; we present theirs w F numbers. The remaining two elements (shown on the lefthand side) are not mixing. We also present the corresponding cumulants κ F . Observe that, among all reduced forests F ∈ F (A), exactly half, presented on top, consists of a single tree (see Remark 3.6).
Definition 1.4. Consider a reduced forest F ∈ F (A). Denote by a v the label of a leaf v. For any vertex v ∈ F we define inductively the quantities κ v as follows: where v 1 , . . . , v n are the descendants of v. For the whole forest F , we define the cumulant κ F to be the product: where V i are the roots of all trees in F (see Figure 1).
Finally, we introduce a class of the mixing forests and the associated quantity w F . Definition 1.5. Let us consider a multiset A = A 1 ∪ · · · ∪ A n and a reduced forest F ∈ F (A). We say that F is mixing for a division A 1 , . . . , A n (or shortly mixing) if for each vertex v whose descendants are all leaves, those descendants are elements of at least two distinct multisets A i and A j . Denote by F(A) the set of all reduced mixing forests.
For a reduced mixing forest F we define the quantity w F to be the number of vertices in F minus the number of leaves (see Figure 1).
We are ready to formulate the main result of this paper.
We briefly present the original formula stated by Leonov and Shiryaev in the framework of an algebra with two multiplications. We use the same notation for multisets A 1 , . . . , A n and its sum A = A 1 ∪ · · · ∪ A n as in Section 1.4.
We introduce a notion of a strongly-mixing partitions (called also indecomposable partitions).
Definition 1.8. Consider a multiset A = A 1 ∪ · · · ∪ A n and any partition ν of A. A partition λ = {λ 1 , λ 2 } is called a row partition if for each multiset A i we have: A partition ν = {ν 1 , . . . , ν q } is called a strongly-mixing partition for the division A = A 1 ∪ · · · ∪ A n (or shortly strongly-mixing partition), if there is no row partition λ such that for any i either ν i ∈ λ 1 , or ν i ∈ λ 2 (see Figure 2).
We denote byP (A) the set of all strongly-mixing partitions of a set A.
We can now express the Leonov-Shiryaev's formula using the notations and notions relevant to the work done in this paper. Theorem 1.9 (Leonov-Shiryaev's formula).
(6) κ a 1 1 · · · a 1 k1 , . . . , a n 1 · · · a n kn = κ where the sum on the right-hand side is running over all strongly-mixing partitions of a set A.

Analogue of Leonov-Shiryaev's formula.
Leonov-Shiryaev's formula relates a cumulant of products with some products of cumulants. In the situation we are interested in this paper, where the conditional expected value is the identity mapping, we can define two types of cumulants. For each of them we have Leonov-Shiryaev's formula. We present now the third formula, which is a mix of those two.
Consider the identity map: between commutative unital algebras (A, ·) and (A, * ). Equation (4) defined cumulants κ of the identity mapping. Observe that we can also consider the inverse mapping, namely the map: . This mapping gives us a way to define cumulants (according to (4)), which we denote by κ * .
We present below the Leonov-Shiryaev's formula for both mappings mentioned above: where the sums in both equalities run over all strongly-mixing partitions of a mul- Observe that in each equality the cumulants on each side are of the same type but the multiplications are not. In our formula we will mix types of cumulants on both sides but keep the same multiplication.
To present our result we introduce a class of strongly-mixing forestsF (A).
Definition 1.11. Let A 1 , . . . , A n be multisets consisting of elements of A. Consider a reduced mixing forest F ∈ F (A) consisting of trees T 1 , . . . , T s . Denote by a v ∈ A the label of a leaf a ∈ A. We define a partition ν F of a set A as follows: We say that a mixing reduced forest F ∈ F (A) is strongly-mixing if the partition ν F is strongly-mixing partition. We denote the set of such forests byF (A). This is analogue to the natural order between classes of strongly-mixing partitionŝ P (A), mixing partitions P (A) and partitions P (A): We can reformulate Theorem 1.6 as follows.
Theorem 1.13 (Analogue of Leonov-Shiryaev's formula). Consider an algebra A with two multiplicative structures · and * . Denote by κ and κ * cumulants related to the identity map on A as we described above. Then the following formula holds: is a set consisting of strongly-mixing reduced forests.
Example 1.14. Figure 1 presents all reduced forests on the multiset A = {a 1 1 , a 1 2 , a 2 1 }. Six of them are mixing, five of them are strongly mixing. Thus: Proof. In (4) we present the moment cumulant formula for cumulants κ related to the map (A, ·) id −→ (A, * ). Similar expression for cumulants κ * related to the is of the following form: We express the ·-product a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn via the moment cumulant formula given by the equation above: a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn = From Theorem 1.6 we can express the left-hand side of this equation in another way: a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn = Observe that we can split the summation of (−1) w F κ F over all mixing reduced forests F ∈ F (A) into * -product of summation over all strongly-mixing reduced forests: Observe, that quantities: satisfy the system of equations given by the moment cumulant formula (8), which has a unique solution. This yields the statement of the theorem.
Remark 1.15. The above equation is still valid when we replace κ (which is hidden in κ F terms) with κ * and replace * -products with ·-products simultaneously.
Observation 1.17. Let us go back to the case, when E is the identity map on algebra A with two multiplications. Suppose that the identity map satisfies the approximate factorization property. Let A 1 , . . . A n be multisets consisting of elements of A. Let A be the sum of those multisets. Then for any forest F ∈ F (A) consisting of f trees, there is the following restriction on the degree of cumulants: where |A| is the number of elements in A.
Proof. We analyse the definition of κ F (Definition 1.4). For any vertex v ∈ F we defined the quantities κ v . Using the approximate factorization property, observe that: Going from the root r to the leaves we obtain: where n r is the number of leaves in a tree rooted in r, and v i for i ∈ [n r ] are leaves of this tree.
The cumulants κ F were defined as follows: It is now easy to see the statement of this observation.
1.8. Application: Jack characters and their structure constants. Jack characters provide dual information about Jack polynomials which are a 'simple' version of Macdonald polynomials [LV95]. Connections of Jack polynomials with various fields of mathematics and physics were established (read more in [Śn15]). Therefore a better understanding of Jack characters might shed some light on Jack polynomials. It seems that behind Jack characters stands a combinatorics of maps, i.e. graphs on the surfaces [DFS13].
Jack characters Ch π form a natural family (indexed by partitions π) of functions on the set Y of Young diagrams. One can introduce two different multiplicative structures on the linear space spanned by Jack characters.
The * -product is given by concatenations of partitions: For any partitions π and σ one can uniquely express the pointwise product of the corresponding Jack characters in the linear basis of Jack characters. The coefficients g µ π,σ (δ) ∈ Q[δ] in this expansion are called structure constants. Each of them is a polynomial in the deformation parameter δ, on which Jack characters depend implicitly. The existence of such polynomials was proven in [DF16]. There are several combinatorial conjectures about structure coefficients [Śn16] and some partial results [KV16,Bur18]. Structure constants are closely related to structure coefficients introduced by Goulden and Jackson in [GJ96].
Śniady considers an algebra of Jack characters as a graded algebra, with gradation given by the notion of α-polynomial functions [Śn15]. Jack characters are α-polynomial function of the following degrees deg Ch π = |π| + ℓ(π).
Śniady gave explicit formulas for the top-degree homogeneous part of Jack characters. We sketch shortly how we use the result presented in the this paper in order to find the top-degree coefficients of the structure constants below. Consider two integer partitions π = (π 1 , . . . , π n ) and σ = (σ 1 , . . . , σ l ) and the relevant multiset A = A 1 ∪ A 2 given by: Together with the ·-product and the * -product described above, the linear space spanned by Jack characters becomes an algebra with two multiplications. We can introduce cumulants as a way of measuring the discrepancy between those two types of multiplications via (4). Recently the approximation factorisation property of cumulants was proven [Śn16].
Proof. Theorem 1.6 presents a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn as a sum over reduced mixing forests of cumulants associated to those forests. Observe that each reduced mixing forest F splits naturally into a collection of trees T 1 , . . . , T k . Each of T i possesses the property of being reduced and mixing. Leaves of F are labelled by elements of A, thus we denoted by A the multiset consisting of those labels.
Theorem 1.19. With notation presented above, for any two partitions π and σ, the following decomposition is valid Moreover, there is the following restriction on the degree of products of cumulants: where |ν| is the number of parts in partition ν.
Presented statement is based on Lemma 1.18, the bound of a degree follows immediately from Observation 1.17.
The division given in (9) is a toll for capturing structure constants g µ π,σ . It opens a way for induction over the number ℓ(σ) + ℓ(π). More precisely, we express κ T in in the linear basis of Jack characters inductively, according to the number of leaves. In a forthcoming paper [Bur18] we give an explicit combinatorial interpretation for the coefficients of high-degree monomials in the deformation parameter δ.
1.9. How to prove the main theorem? Theorem 1.6 is a straightforward conclusion from two propositions which we present in this section. In our opinion they are interesting themselves.
We begin by introducing a gap-free vertex colouring on forests F ∈ F (A).
Definition 1.20. For a reduced forest F with leaves in a multiset A = A 1 ∪ · · · ∪ A n we say that c is a gap-free vertex colouring with length r if • c is a coloured by the numbers {0, . . . , r} and each colour is used at least once; • each leaf is coloured by 0; • the colours are strictly increasing on any path from the root to a leaf. We denote by |c| := r the length of c . We call such a colouring c weakly-mixing if it satisfies one of the following additional conditions: (1) either there exists a vertex coloured by 1 with at least two descendants, each of whom belongs to a distinct multiset A i , (2) or colouring c does not use the colour 1 at all.
We denote by C F the set of all gap-free and weakly-mixing colourings of a forest F .
The following result is a juggling of a concept of cumulants. We present its proof in Section 2. A 1 , . . . , A n be multisets consisting of elements of A. Let A be a sum of those multisets. Then (10) a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn =

Proposition 1.21. Let
In Section 3, we will show that summing over all colourings c ∈ C F for a reduced forest F ∈ F (A) gives a surprisingly simple number. This result is presented in proposition below. Proposition 1.22. Let A 1 , . . . , A n be multisets consisting of elements of A. Let A be a sum of those multisets. Then, for any reduced forest F ∈ F (A), the following holds: otherwise.
Observe that combing Proposition 1.21 and Proposition 1.22 we obtain the statement of Theorem 1.6.

Related work and free probability theory.
A noncommutative probability space is a pair (A, φ) consisting of a unital algebra A and a linear form φ on A such that φ(1) = 1, which is called a noncommutative expectation [Spe94,Spe98]. Functions m n (a 1 , . . . , a n ) := φ (a 1 · · · a n ) are called free moments.
Roland Speicher introduced the free cumulant functional [Spe94] in the free probability theory. It is related to the lattice of noncrossing partitions of the set [n] in the same way in which the classic cumulant functional is related to the lattice of all partitions of that set.
Definition 1.23. A partition ν ∈ P([n]) is noncrossing if there is no quadruple of elements i < j < k < l such that i ∼ ν k, j ∼ ν l, and ¬(i ∼ ν j), where "∼ ν " denotes the relation of being in the same set in the partition ν.
The free cumulants are defined implicit by the system of equations φ (a 1 · · · a n ) = where the sum runs over all noncrossing set partitions of [n], compare to (3). Möbius inversion over the lattice of noncrossing partitions gives the formula for the free cumulants in terms of the free moments.
Josuat-Vergès, Menous, Novelli and Thibon [JVMNT17, Theorem 4.2] give formulas for free cumulants in terms of Schröder trees, i.e. reduced plane trees for which the rightmost sub-tree is a leaf. To each such a tree they associate the term constructed by the mapping φ. At the first glance the formula seems to be related to the formula we give in a current paper. However, the reasons for appearance of reduced trees in both papers are different. In their work reducedness of trees is a natural property appearing while recovering the free cumulant from free moments. In our case, where E or φ is the identity, we have κ (a) = a for any element a ∈ A. Hence we consider reduced trees. The notion of free cumulants is based on noncrossing partitions which follows flatness of trees they consider.
There are approaches to freeness other then considering a linear form φ on an algebra A. The standard and the most common approach is to additionally require that A is a B-module or B ⊆ A is a subalgebra of A. The mapping φ : A −→ B satisfies the bimodule map property: for any a ∈ A and b 1 , b 2 ∈ B. There are several slightly different approaches, e.g., free products came with amalgamation over module B [Voi95], where cumulants and moments are operator-valued multiplicative functions [Spe98, Definition 2.1.1].
Our work is based on the idea of taking an identity map between elements of an algebra with two different multiplications (φ ≡ id). Such situation does not arise naturally in free probability theory, where φ is usually either linear form or bimodule map on an algebra A. It rises the question if it is still possible to define naturally cumulants in the setup of noncommutative algebras with two different products.

Proof of Proposition 1.21
In this section we shall prove Proposition 1.21. We use the same notation as in Section 1.4. We denote by A 1 , . . . , A n multisets consisting of elements of A. We denote by A = A 1 ∪ · · · ∪ A n the multiset, which is the sum of all multisets A i . We use also the following notation for the elements of A i : . . , a i ki , hence the multiset A consists of the following elements: a 1  1 , . . . , a 1  k1 , . . . , a n 1 , . . . , a n kn .
We denote additionally the set of all partitions of A by P (A). We denote by P (A) a set of all mixing partitions of A, i.e. all partitions ν = {ν 1 , . . . , ν l } such that 2.1. Outline of the proof. Firstly, we express the left-hand side of (10) as a sum of cumulants, where the sum runs over all mixing partitions ν ∈ P (A), see (11) below. By applying inductively the procedure (12) described below, we replace summation over all mixing partitions ν ∈ P (A) by a sum over all nested upward sequences of partitions, see Definition 2.2. Then we construct a bijection between such sequences and reduced forests F ∈ F (A) equipped with gap-free, weaklymixing colourings c ∈ C F (see Definitions 1.3, 1.20). Later on we will prove that the weighted sum over all gap-free colourings for a fixed forest is either equal to 0 or to ±1.

Cumulants of upward sequences of partitions.
Each cumulant on the right-hand side of (11) is a ·-product of simple cumulants. We use the momentcumulant formula in a form given below a 1 , . . . , a k ).
to replace ·-products by * -products and ·-products consisting of a strictly smaller number of components.
For each cumulant on the right-hand side in (11) we apply the procedure (12). As an output we get one term which is a * -product of cumulants and several terms of the form of a ·-product of cumulants. Observe that in each term of the second type the number of factors is strictly smaller than before applying the procedure. We apply to them this procedure iteratively as long as we have ·-terms in our extension. In the end we get a sum of the terms given by * -product and cumulants.
Definition 2.2. A sequence of partitions ω = ν 1 ր · · · ր ν r is said to be upward if ν i+1 is a partition of the set ν i , for any 1 ≤ i ≤ r − 1 and ν 1 is a partition of a multiset A. Moreover, if for each i the partition ν i+1 is non-trivial, i.e. ν i+1 = ν i , it is said to be nested. We define the length of an upward sequence of partitions ω = ν 1 ր · · · ր ν r as the length of a sequence, and we denote |ω| = r.
Let us provide a simple example.
Definition 2.6. Consider a multiset A = A 1 ∪· · ·∪A n . Denote by N (A) the set of all nested upward sequences of partitions ω = ν 1 ր · · · ր ν r such that ν 1 ∈ P (A) is a mixing partition.

Proposition 2.7. Consider a multiset
Proof. Apply procedure (12) iteratively to the left-hand side of Proposition 2.7. Observe that applying this iterative procedure is nothing else but summing over all nested upward sequences of partitions ω = ν 1 ր · · · ր ν r . The sign of the term is determined by the number of iterations. Partition ν 1 describes the first application of the procedure (this is why ν 1 ∈ P (A)), partition ν 2 the second, and so on.
Observe that different nested upward sequences of partitions ω may lead to the same cumulant κ ω . The following example illustrates this phenomenon.

Reduced forests and their colourings.
To each upward nested sequence of partitions ω = ν 1 ր · · · ր ν r we shall assign a certain rooted forest with a colouring. We construct a bijection between the sequences from N (A) and relevant rooted forests equipped with the colourings.
• For each element ν i j , where 1 ≤ i ≤ r and 1 ≤ j ≤ k i , we create a vertex and colour it by i.
Similarly we join a ∈ A and ν 1 j if a ∈ ν 1 j . • We delete each vertex v which has only one descendant. We join the descendant and the parent of v. We denote by Φ 1 (ω) the forest and by Φ 2 (ω) the colouring associated to ω.
The forest described in Definition 2.9 consists of k r rooted trees, where k r is a number of elements in ν r , namely ν r = {ν r 1 , . . . , ν r kr }. The condition of nestedness of ω translates to the fact that each colour is used. Except for leaves, each vertex has at least two descendants. It leads to the definition of reduced forest and gapfree, weakly-mixing colouring. We mentioned their definitions in the introduction (see Definitions 1.3 and 1.20). (F, c) consisting of a reduced forest F ∈ F (A) of length r ≥ 1 with a gap-free, weakly-mixing colouring c ∈ C F :

Lemma 2.11. There exists a bijection Φ between the set N (A) of nested upward sequences starting with a mixing partition and the set of pairs
For any nested upward sequence starting with a mixing partition ω ∈ N (A), the following equality of cumulants holds where κ Φ1(ω) is a cumulant of a reduced forest Φ 1 (ω), see Definition 1.4.
Moreover |ω| = |Φ 2 (ω)| i.e. the length of the nested upward sequence is equal to the length of the corresponding colouring.
Proof. Definition 2.9 shows already how to associate a reduced forest F := Φ 1 (ω) with the gap-free colouring c := Φ 2 (ω) to a nested upward sequence ω. The construction is done in such a way that |c| = |ω|. For the reverse direction, the algorithm is easily reproducible. The condition that a nested upward sequence ω = ν 1 ր · · · ր ν r ∈ N (A) starts with a mixing partition ν 1 ∈ P (A) translates to the condition of c being a weakly-mixing colouring (Definition 1.20).
In Definition 1.4 we introduced cumulant κ F for a forest F ∈ F (A). There is an exact correspondence between this expression and the one, which is given in Definition 2.4.
We are ready to prove Proposition 1.21 which is the purpose of this section. Let us recall its statement: Proposition 1.21. Let A 1 , . . . , A n be multisets consisting of elements of A. Let A be a sum of those multisets. Then (10) a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn = Proof. Combining the formula (11) and the Proposition 2.7 lead to the following expression: a 1 1 * · · · * a 1 k1 · · · a n 1 * · · · * a n kn = a 1 1 * · · · * a 1 k1 * · · · * a n 1 * · · · * a n kn + We identify the product term on the right-hand side of the equation above with the only reduced forest of length r = 0. Indeed, there is just one reduced forest of length r = 0 and the only one gap-free, weakly-mixing vertex colouring c of it, namely the forest F consisting of separated vertices a ∈ A, each coloured by 0. The term a 1 1 * · · · * a 1 k1 * · · · * a n 1 * · · · * a n kn is equal to the corresponding cumulant κ F .
We replace the sum term on the right-hand side of the equation above, according to the bijection between sequences ω ∈ N (A) and reduced forests of length r ≥ 1 with gap-free, weakly-mixing colourings given in Lemma 2.11.

Proof of Proposition 1.22
In this section we shall prove Proposition 1.22. For a given reduced forest F , we investigate the following sum c∈CF (−1) |c| over all gap-free, weakly-mixing colourings of F , which occur in Proposition 1.21.
3.1. Parameter w F of a reduced forest F ∈ F (A). We introduce an invariant w F which determines the coefficient of κ F . This definition was already mentioned in Section 1.4, we recall it below and next extend it slightly: Definition 1.5. Let us consider a multiset A = A 1 ∪ · · · ∪ A n and a reduced forest F ∈ F (A). We say that F is mixing for a division A 1 , . . . , A n (or shortly mixing) if for each vertex v whose descendants are all leaves, those descendants are elements of at least two distinct multisets A i and A j . Denote by F(A) the set of all reduced mixing forests.
For a reduced mixing forest F we define the quantity w F to be the number of vertices in F minus the number of leaves (see Figure 1).
If F is not mixing, we define w F := ∞. We may also introduce number w F inductively, according to the height of a forest.
Definition 3.1. Let T be a reduced tree. The height of a tree T is the maximum distance between its root and one of its leaves. The hight of a forest F is a hight of the highest tree in F . We denote this quantity by h(F ).
Definition 3.2 (Definition equivalent to Definition 1.5). Let F be a reduced forest and F 1 , . . . , F r its sub-forests obtained by deleting the roots of F . We define the number w F ∈ N ∪ {∞} inductively on h(F ) as follows: if h(F ) = 1 and all descendants of some root belong to just one multiset A i for some i ∈ [n], 1 if h(F ) = 1 and for each root there are at least two descendants belonging to some two distinct multisets A i and A j , 0 if h(F ) = 0.
Example 3.3. Let A = A 1 ∪A 2 ∪A 3 . On Figure 4, we give an example of two forests (in particular trees) and we count the two corresponding w F numbers. Observe that number w F depends on the labels of the leaves of F .

The proof of Proposition 1.22.
Let us recall the statement of proposition. A 1 , . . . , A n be multisets consisting of elements of A. Let A be a sum of those multisets. Then, for any reduced forest F ∈ F (A), the following holds:
The proof of Proposition 1.22 is divided into two cases: either a forest F is not mixing, i.e. w F = ∞ (Lemma 3.4), or a forest F is mixing, i.e. w F = ∞ (Lemma 3.5). The next two subsections establish these two cases. Proof. Since F is not mixing, there exists a vertex v such that all of its descendants are leaves, and all of them belong to just one multiset A i for some i ∈ [n]. Consider the following partition of set C F : where each C 1 i consists of all c ∈ C F with |c| = i and where the vertex v is coloured by its own colour; C 2 i consists of all c ∈ C F with |c| = i and where there is another vertex coloured by the same colour as the vertex v. We express the sum over all c ∈ C F as follows: We will show the equipotency of the sets C 1 i and C 2 i−1 from which it follows that the sum above is equal to 0 and the statement of the lemma is true.
Let us construct a bijection between C 1 i and C 2 i−1 . Take any c ∈ C 1 i . Suppose that the vertex v is coloured by k. Observe that k ≥ 2. Indeed, if v was coloured by 1, it would be the only vertex of this colour. Then, the only 1-coloured vertex would have descendants belonging to just one multiset A i , which is in contradiction with the fact that c ∈ C F (i.e. c is a weakly-mixing colouring). From c ∈ C 1 i we construct c ′ ∈ C 1 i−1 as follows: (1) keep the colours of vertices coloured by 1, . . . , k − 1 unchanged, (2) change the colours of vertices coloured by k, . . . , i to k − 1, . . . , i − 1 respectively.
This procedure is reversible. Indeed, take c ′ ∈ C 1 i−1 and suppose that vertex v is coloured by k for some k ≥ 1. Then c ∈ C 2 i can be recovered by the following procedure: (1) do not change colours of the vertices coloured by 1, . . . , k − 1, (2) do not change the colour of v, (3) change the colours of the vertices coloured by k, . . . , i − 1 to k + 1, . . . , i respectively (excluding vertex v).

How to prove the mixing case?
We will prove the following lemma. To prove the lemma above we show a bijection between gap-free colourings of reduced trees T ∈ T (A) and gap-free colourings of reduced forests F ∈ F (A), which are not trees (see Remark 3.6). Using this bijection we can restrict the proof of Lemma 3.5 just to trees. For reduced trees and their gap-free colourings we define a projection of this colourings (see Definition 3.7). We make use of the notion of projection in Lemma 3.8. Proof of Lemma 3.5 is done by induction on number of vertices in tree T and presented in Section 3.7.

Restriction to the trees.
Remark 3.6. There is a natural bijection f between all reduced trees T ∈ T (A) and all reduced forests F ∈ F (A), which are not trees. This bijection is obtained by deleting the root of T (see Figure 1). Moreover, for a given reduced tree T ∈ T (A), there is an obvious bijection f T between all gap-free colourings of T and all gapfree colourings of the corresponding reduced forest f (T ), obtained by keeping the colours of the non-deleted vertices, so that |f T (c)| = |c| − 1.
Additionally f T preserves the property of being a weakly-mixing colouring.
The above statement allows us to prove Lemma 3.5 just for the case of trees T ∈ T (A) and conclude the statement for all forests F ∈ F (A). Indeed, suppose that the statement of Lemma 3.5 holds for trees. Consider a mixing forest F ∈ F (A) which is not a tree. Then, the tree T := f −1 (F ) ∈ T (A) is also mixing, hence we can use the statement of Lemma 3.5. Observe that w F = w T − 1. Using bijections f and f T we get the following equality which is the statement of Lemma 3.5 for the mixing forest F ∈ F (A).  By deleting the root we obtain two reduced subtrees: T 1 and T 2 with inherited colourings:c 1 andc 2 . Observe that they are not gapfree. (c) However, the procedure given in Definition 3.7 describes the canonical way of producing a gap-free colourings p 1 (c) and p 2 (c). (d) Moreover, for the colouring c we present an associated path ρ which will be introduced in a proof of Lemma 3.8.

Projection of a gap-free colouring.
For any reduced tree T ∈ T (A) we consider sub-trees T 1 , . . . , T k formed by deleting the root of T . Number k is equal to the degree of the root. Every sub-tree T i is also a reduced tree. Every gap-free colouring c induces also a sub-colouringsc 1 , . . . ,c k on T 1 , . . . , T k . Observe that sub-colourings obtained this way are not necessarily gap-free. However, there is a canonical way to make them gap-free.
Definition 3.7. Let T be a reduced tree with a gap-free colouring c. Letc 1 , . . . ,c k be the induced colourings on sub-trees T 1 , . . . , T k formed by deleting the root of T . For some i ∈ [k], let j i 0 < · · · < j i l be the sequence of colours used in the colourinḡ c i . By replacing each j i n by n in the colouringc i we obtain a gap-free colouring, which we denote by c i . We say that c i as an i-th projection of the colouring c and denote it as p i (c) := c i , see Figure 5.  Let c 1 , . . . , c r be gap-free colourings of T 1 , . . . , T r respectively. Then the following equality holds: Proof of Lemma 3.8. Observe that if T is the mixing tree, then any gap-free colouring c belongs to C T . Indeed, take any vertex v coloured by 1. Clearly, its descendants are leaves labelled by elements of at least two distinct multisets A i and A j (by assumption that T is mixing). The existence of such a vertex implies that c ∈ C T . The proof is divided into three steps: first, we construct a bijection between the gap-free colouring c projecting onto c 1 , . . . , c r and some integer paths in N r ; second, we introduce a generating function of these paths and characterise it by recursion on the endpoints and some boundary condition; finally, we find a function satisfying those conditions.
Step 1. Let us recall that any gap-free colouring c of T induces colourings c i on T i , from which we deduce a gap-free colouring c i of T i (Definition 3.7). We shall construct a bijection between all gap-free colourings c of T projecting on c 1 , . . . , c r and all integer paths ρ such that: • ρ connects (0, . . . , 0) and (|c 1 |, . . . , |c r |) ∈ N r , • each step of ρ is of the following form: (k n 1 , . . . , k n r ) ∈ {0, 1} r \ (0, . . . , 0) . Denote the class of such paths by P |c1|,...,|cr| . Moreover the construction is done in such a way that |c| = |ρ| + 1, where by |ρ| we denote the number of steps in ρ.
For a gap-free colouring c we construct a path ρ starting from (0, . . . , 0) ∈ N r by the following procedure: the n-th step of ρ is of the form (k n 1 , . . . , k n r ) where: if colour n appears in c i , 0 if it does not.
An example of such path is presented on Figure 5. The procedure described above is reversible. Indeed, take a path ρ between (0, . . . , 0) and (|c 1 |, . . . , |c r |) ∈ N r . Suppose that the n-th step is of the form: (k n 1 , . . . , k n r ) ∈ {0, 1} r \ (0, . . . , 0) . We can assign to the path ρ a colouring c by the following procedure. Let (x 1 , . . . , x r ) be an endpoint of ρ after the n-th step. We colour each vertex v ∈ F i by n if v was coloured by x i in colouring c i and k n i = 0. We colour the root by |ρ| + 1.
Step 2. The bijection from Step 1 was constructed in such a way that |c| = |ρ|+1. Observe that Let us define a function F : Z r −→ Z: Observe that: I. for all (x 1 , . . . , x r ) ∈ N r , F (x 1 , . . . , x r ) = 0, II. F (0, . . . , 0) = −1, III. for all (x 1 , . . . , x r ) ∈ N r \ (0, . . . , 0), the function F satisfies the following recursive formula: Let us shortly comment on this observation. There are no paths connecting (0, . . . , 0) with points (x 1 , . . . , x r ) ∈ N r using the set of steps which is non-negative (Observation I). There is just one path connecting point (0, . . . , 0) to itself: the empty path. Its length is equal to 0 (Observation II). Consider all possibilities for the last step in path ρ. It is equivalent to choosing indices X ⊂ [r], X = ∅ and summing over all paths ending in (x X 1 , . . . , x X r ) multiplied by −1, because we count the sign of the path (Observation III).
Those three statements about function F define it uniquely. The recursive formula gives us the way to compute F (x 1 , . . . , x r ) inductively according to x i = 0.
Step 3. We show now that the function G : Z r −→ Z: We shall show that function G satisfies three properties I, II, III. Clearly, it satisfies I, II. In order to show that the recursive formula also holds, take any Observe that summants of the sum over X ⊂ Y are equal to 0. From X ⊂ Y it follow that there exists i ∈ [r] such that i ∈ X and i ∈ Y . It means that x X i = −1 and by definition G(x X 1 , . . . , x X r ) = 0. Observe that the summants in the sum over X ⊂ Y are of the form − r i=1 (−1) x X i . Indeed, from X ⊂ Y it follows that x X i ≥ 0 for all i ∈ [r]. Thus we have

Proof of Lemma 3.5.
Remark 3.9. Let T be a reduced mixing tree such that h(T ) ≤ 2. Let T 1 , . . . , T r be sub-trees obtained from F by deleting the roots. For any colouring c ∈ C F the projection c i := p i (c) is in C Ti for each i ∈ [r]. Indeed, by definition, c i = p i (c) are gap-free colourings of F i . Take any vertex v coloured by 1 in c i . Its descendants are leaves. Hence w F = ∞, so also k Fi = ∞. That means that descendants of v belong to at least two distinct multisets A i . The existence of such vertex implies that c ∈ C Ti .
Proof of Proposition 3.5. We will use induction on the height of tree T ∈ T (A).
We cannot begin with a tree of height 0, namely a one-vertex tree T = •, because there is no such s tree if |A| ≥ 2.
Induction base. We begin from a tree T of height one, namely consisting just of the root and leaves. We have exactly one gap-free colouring of length 1, namely leaves are coloured by 0 and the root by 1. The claim follows immediately.
Induction step. Let n ≥ 2. Suppose now that the statement of Lemma 3.5 is true for any tree T of the height h(T ) ≤ n − 1. We will show that it is also true for any tree of height equal to n ≥ 2. Take such a tree T . Denote by T 1 , . . . , T r its sub-trees obtained from T by deleting the root. Clearly for every T i , the h(T i ) ≤ n − 1 and we can use the induction hypothesis for them. We have: which proves the statement for any tree of height equal to n.