Method for Obtaining Coefficients of Powers of Multivariate Generating Functions

Abstract

There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is the application of compositae of generating functions. In previous studies, we have introduced a generalization for the compositae of multivariate generating functions and have defined basic operations on the compositae of bivariate generating functions. The use of these operations helps to obtain explicit formulas for compositae and coefficients of generating functions in two variables. In this paper, we expand these operations on compositae to the case of generating functions in three variables. In addition, we describe a way of applying compositae to obtain coefficients of rational generating functions in several variables. To confirm the effectiveness of using the proposed method, we present detailed examples of its application in obtaining explicit formulas for the coefficients of a generating function related to the Aztec diamond and a generating function related to the permutations with cycles.

1. Introduction

Generating functions find their application in various areas of mathematics and computer science, such as combinatorics, discrete mathematics, statistics, etc. [1,2]. Generally, generating functions have an important application to enumeration problems related to counting different kinds of objects. For example, a compact representation of discrete structures with the help of generating functions is used in combinatorics [3,4]. In this case, the coefficients of the known generating function for a combinatorial set show the total number of its elements (combinatorial objects). Thus, the explicit formula for the generating function coefficients can be used as a cardinality function of the corresponding combinatorial set. Moreover, generating functions are the main means of describing polynomials that can be found in different tasks of mathematics [5,6,7]. In this case, the coefficients of the known generating function for a polynomial define the expression of this polynomial with the fixed values of its parameters. Hence, it is necessary to have methods for obtaining explicit formulas or recurrences for the coefficients of generating functions. It is also useful to have methods for obtaining recurrences for the coefficients of generating functions.

When studying the coefficients of generating functions, special attention is paid to the powers of these generating functions. There are general concepts for obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers:

• Potential polynomials introduced by Comtet [8];
• Riordan arrays introduced by Shapiro et al. [9];
• Power matrices introduced by Knuth [10];
• Compositae introduced by Kruchinin [11,12].

However, all these concepts are based on the processing of generating functions in one variable. At the same time, there are many problems where the processing of generating functions in several variables (multivariate generating functions) is required. Let us consider some papers that can be cited as examples of such studies, which are based on the use of generating functions. Generating functions are actively used in the study of various lattice paths. For instance, the multivariate generating functions for four classes of lattice paths (walks, bridges, excursions, and excursions) were obtained in [13]. The study of generating functions for the class of walks with small steps in the quarter plane is presented in [14]. There are also works that connect generating functions with other classes of discrete structures, for example: a multivariate generating function for all semi-magic squares was obtained in [15]; a multivariate generating function for information spread on multi-type random graphs was obtained in [16]. Another task related to the development of algorithms for solving a multidimesional difference equation with constant, polynomial, or rational function coefficients can be found in [17]. There are also several studies on the multivariate case of the relationship between the generating function of a solution to a linear Cauchy problem [18,19,20]. These papers are only a small part of the research related to multivariate generating functions and confirm the relevance of the development of methods for processing such generating functions.

The process of obtaining coefficients of multivariate generating functions was studied in more detail by Pemantle et al. [21,22]. However, they considered only asymptotic methods for solving this problem.

To obtain coefficients of multivariate generating functions explicitly, we consider the concept of compositae introduced in our previous studies [23].

Definition 1. 

The composita $F^{\Delta }(n,m,\dots ,l,k)$ of a multivariate generating function

$$F(x,y,\dots ,z)=\sum _{n\ge 0}\sum _{m\ge 0}\dots \sum _{l\ge 0}f(n,m,\dots ,l)x^n{y}^m\cdots z^l,ord\left(F\right)\ge 1,$$

is a coefficient’s function of the kth power of the generating function $F(x,y,\dots ,z)$:

$$F{(x,y,\dots ,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\dots \sum _{l\ge 0}{F}^{\Delta }(n,m,\dots ,l,k)x^n{y}^m\cdots z^l,$$

where $F{(x,y,\dots ,z)}^0=1$.

In this case, the following definition of the order of a formal power series $F(x,y,\dots ,z)$ is used [24]:

$$ord\left(F\right)=\left\{\begin{array}{cc}min\{r=n+m+\dots +l:f(n,m,\dots ,l)\ne 0\},\hfill & \mathrm{if}F(x,y,\dots ,z)\ne 0;\hfill \\ +\infty ,\hfill & \mathrm{if}F(x,y,\dots ,z)=0.\hfill \end{array}\right.$$

By applying basic operations on the compositae (composition, addition, multiplication, reciprocation, and compositional inversion), it is possible to obtain explicit formulas for compositae and coefficients of generating functions in two variables (bivariate generating functions). In this paper, we expand these operations on compositae to the case of multivariate generating functions and present an example of their application to a generating function in three variables.

The organization of this paper is as follows. In Section 2, we introduce the following basic operations on compositae of generating functions in three variables: composition, addition, and multiplication. Then, in Section 3, we consider a general scheme for obtaining the coefficients of a rational generating function in n variables. To confirm the effectiveness of using compositae, we present examples of applying the proposed method and obtain explicit formulas for the coefficients of generating functions in three variables. The obtained results are shown in Section 4.

2. Operations on Compositae of Generating Functions in Three Variables

In this section, we consider an extension of the method based on the use of compositae to the case of generating functions in three variables.

A generating function in three variables is a formal power series of the following form:

$$F(x,y,z)=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}f(n,m,l)x^n{y}^m{z}^l.$$

The k-th power of the generating function $F(x,y,z)$, with $ord\left(F\right)\ge 1$, based on the concept of compositae, can be presented as follows:

$$F{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{F}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l.$$

Next, we introduce and prove the following basic operations on compositae of generating functions in three variables: composition, addition, and multiplication. The use of these operations helps to obtain explicit formulas for compositae and coefficients of generating functions in three variables.

2.1. Composition of Generating Functions in Three Variables

First, let us consider the process of obtaining an explicit formula for the coefficients of the generating function in three variables that is the result of the composition of generating functions in three variables.

Theorem 1. 

Suppose that:

$$H(x,y,z)=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}h(n,m,l)x^n{y}^m{z}^l,$$

$$A{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{A}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(A\right)\ge 1,$$

$$B{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{B}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(B\right)\ge 1,$$

$$C{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{C}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(C\right)\ge 1.$$

Then, the coefficients $g(n,m,l)$ of the composition of generating functions in three variables

$$G(x,y,z)=H(A(x,y,z),B(x,y,z),C(x,y,z))=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}g(n,m,l)x^n{y}^m{z}^l$$

are equal to

$$g(n,m,l)=\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}h(k_a,k_b,k_c)·h_{ABC}(n,m,l,k_a,k_b,k_c),$$

(1)

where

$$h_{ABC}(n,m,l,k_a,k_b,k_c)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{h}_{AB}(n_i,m_i,l_i,k_a,k_b)·C^{\Delta }(n-n_i,m-m_i,l-l_i,k_c),$$

(2)

$$h_{AB}(n,m,l,k_a,k_b)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{A}^{\Delta }(n_i,m_i,l_i,k_a)·B^{\Delta }(n-n_i,m-m_i,l-l_i,k_b).$$

(3)

Proof. 

Let us expand the given composition of generating functions in three variables and obtain

$$G(x,y,z)=H\left(A\right(x,y,z),B(x,y,z),C(x,y,z\left)\right)=$$

$$=\sum _{k_a\ge 0}\sum _{k_b\ge 0}\sum _{k_c\ge 0}h(k_a,k_b,k_c)·A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}·C{(x,y,z)}^{k_c}.$$

The end part of this expression can be written as follows:

$$H_{ABC}(x,y,z)=A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}·C{(x,y,z)}^{k_c}=$$

$$=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{h}_{ABC}(n,m,l,k_a,k_b,k_c)x^n{y}^m{z}^l.$$

Combining these formulas, we have

$$G(x,y,z)=\sum _{k_a\ge 0}\sum _{k_b\ge 0}\sum _{k_c\ge 0}\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}h(k_a,k_b,k_c)·h_{ABC}(n,m,l,k_a,k_b,k_c)x^n{y}^m{z}^l$$

and

$$g(n,m,k)=\sum _{k_a\ge 0}\sum _{k_b\ge 0}\sum _{k_c\ge 0}h(k_a,k_b,k_c)·h_{ABC}(n,m,l,k_a,k_b,k_c).$$

(4)

We also represent $H_{ABC}(x,y,z)$ as follows:

$$H_{ABC}(x,y,z)=H_{AB}(x,y,z)·C{(x,y,z)}^{k_c},$$

where

$$H_{AB}(x,y,z)=A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{h}_{AB}(n,m,l,k_a,k_b)x^n{y}^m{z}^l.$$

The coefficients of the product of two generating functions can be obtained using the convolution operation [25]:

• To obtain an explicit formula for the coefficients $h_{AB}(n,m,l,k_a,k_b)$, we apply the convolution operation to

  $$H_{AB}(x,y,z)=A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}$$

  and obtain Equation (3).
• To obtain an explicit formula for the coefficients $h_{ABC}(n,m,l,k_a,k_b,k_c)$, we apply the convolution operation to

  $$H_{ABC}(x,y,z)=H_{AB}(x,y,z)·C{(x,y,z)}^{k_c}$$

  and obtain Equation (2).

Since $ord\left(A\right)\ge 1$, $ord\left(B\right)\ge 1$ and $ord\left(C\right)\ge 1$, then for $H_{ABC}(x,y,z)$, we have

$$ord\left(H_{ABC}(x,y,z)\right)=ord(A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}·C{(x,y,z)}^{k_c})\ge k_a+k_b+k_c.$$

Thus, we have the following restriction for the indices of summation when calculating $g(n,m,l)$ in Equation (4):

$$k_a+k_b+k_c\ge n+m+l.$$

Combining all the obtained formulas, we derive the desired result that the coefficients $g(n,m,l)$ of the composition of generating functions in three variables can be calculated by Equation (1). □

We also consider the process of obtaining an explicit formula for the composita of the generating function in three variables $G(x,y,z)$.

Theorem 2. 

Suppose that:

$$H{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{H}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(H\right)\ge 1,$$

$$A{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{A}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(A\right)\ge 1,$$

$$B{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{B}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(B\right)\ge 1,$$

$$C{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{C}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(C\right)\ge 1.$$

Then, the composita $G^{\Delta }(n,m,l,k)$ of the composition of generating functions in three variables

$$G(x,y,z)=H(A(x,y,z),B(x,y,z),C(x,y,z))=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}g(n,m,l)x^n{y}^m{z}^l$$

is equal to

$$G^{\Delta }(n,m,l,k)=\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}{H}^{\Delta }(k_a,k_b,k_c,k)·h_{ABC}(n,m,l,k_a,k_b,k_c),$$

where

$$h_{ABC}(n,m,l,k_a,k_b,k_c)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{h}_{AB}(n_i,m_i,l_i,k_a,k_b)·C^{\Delta }(n-n_i,m-m_i,l-l_i,k_c),$$

$$h_{AB}(n,m,l,k_a,k_b)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{A}^{\Delta }(n_i,m_i,l_i,k_a)·B^{\Delta }(n-n_i,m-m_i,l-l_i,k_b).$$

Proof. 

The proof of Theorem 2 is similar to the proof of Theorem 1. □

2.2. Addition of Generating Functions in Three Variables

Next, let us consider the process of obtaining an explicit formula for the composita of the generating function in three variables that is the result of the addition of two generating functions in three variables.

Theorem 3. 

Suppose that:

$$A{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{A}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(A\right)\ge 1,$$

$$B{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{B}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(B\right)\ge 1.$$

Then, the composita $G^{\Delta }(n,m,l,k)$ of the addition of generating functions in three variables

$$G(x,y,z)=A(x,y,z)+B(x,y,z)=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}g(n,m,l)x^n{y}^m{z}^l$$

is equal to

$$G^{\Delta }(n,m,l,k)=$$

(5)

$$=\sum _{k_a=0}^{n+m+l}\left(\genfrac{}{}{0pt}{}{k}{k_a}\right)\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{A}^{\Delta }(n_i,m_i,l_i,k_a)·B^{\Delta }(n-n_i,m-m_i,l-l_i,k-k_a).$$

Proof. 

For the bivariate generating function

$$H(x,y)=x+y,$$

its k-th power can be obtained by appying the binomial theorem:

$$H{(x,y)}^k={(x+y)}^k=\sum _{n\ge 0}\sum _{m\ge 0}{H}^{\Delta }(n,m,k)x^n{y}^m=\sum _{n\ge 0}\sum _{m\ge 0}\left(\genfrac{}{}{0pt}{}{k}{n}\right)\delta (m,k-n)x^n{y}^m.$$

In this expression, we use the Kronecker delta function

$$\delta (i,j)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}i\ne j;\hfill \\ 1,\hfill & \mathrm{if}i=j.\hfill \end{array}\right.$$

Hence, the composita of the generating function $H(x,y)$ is

$$H^{\Delta }(n,m,k)=\left(\genfrac{}{}{0pt}{}{k}{n}\right)\delta (m,k-n).$$

Next, we represent the generating function $H(x,y)$ as the following generating function in three variables:

$$H_{xy}(x,y,z)=H(x,y)z^0=H(x,y).$$

Hence, the composita of the generating function $H_{xy}(x,y,z)$ is

$$H_{xy}^{\Delta }(n,m,l,k)=H^{\Delta }(n,m,k)\delta (l,0)=\left(\genfrac{}{}{0pt}{}{k}{n}\right)\delta (m,k-n)\delta (l,0).$$

We also consider a generating function in three variables

$$C(x,y,z)=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}c(n,m,l)x^n{y}^m{z}^l,ord\left(B\right)\ge 1,$$

$$C{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{C}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l.$$

Applying Theorem 2 for the composition of generating functions

$$G(x,y,z)=A(x,y,z)+B(x,y,z)=H_{xy}(A(x,y,z),B(x,y,z),C(x,y,z)),$$

we obtain

$$G^{\Delta }(n,m,l,k)=$$

$$=\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}{H}_{xy}^{\Delta }(k_a,k_b,k_c,k)·h_{ABC}(n,m,l,k_a,k_b,k_c)=$$

$$=\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}\left(\genfrac{}{}{0pt}{}{k}{k_a}\right)\delta (k_b,k-k_a)\delta (k_c,0)h_{ABC}(n,m,l,k_a,k_b,k_c),$$

where $h_{ABC}(n,m,l,k_a,k_b,k_c)$ are the coefficients of the generating function

$$H_{ABC}(x,y,z)=A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}·C{(x,y,z)}^{k_c}.$$

Simplifying expressions of the Kronecker delta function, we obtain $k_c=0$ and $k_b=k-k_a$.

Therefore, we have

$$G^{\Delta }(n,m,l,k)=\sum _{k_a=0}^{n+m+l}\left(\genfrac{}{}{0pt}{}{k}{k_a}\right)h_{ABC}(n,m,l,k_a,k-k_a,0).$$

Applying Equation (2) for $h_{ABC}(n,m,l,k_a,k-k_a,0)$, we obtain

$$h_{ABC}(n,m,l,k_a,k-k_a,0)=$$

$$=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{h}_{AB}(n_i,m_i,l_i,k_a,k-k_a)·C^{\Delta }(n-n_i,m-m_i,l-l_i,0),$$

where $h_{AB}(n,m,l,k_a,k_b)$ are the coefficients of the generating function

$$H_{AB}(x,y,z)=A{(x,y,z)}^{k_a}·B{(x,y,z)}^{k_b}.$$

According to the initial condition for the compositae of multivariate generating functions [23], the value of the composita $C^{\Delta }(n-n_i,m-m_i,l-l_i,0)$ is equal to 1 only when $n-n_i=0$, $m-m_i=0$ and $l-l_i=0$; otherwise, it is equal to 0. Thus, we obtain $n_i=n$, $m_i=m$, and $l_i=l$. In this case, we derive the following formula for $h_{ABC}(n,m,l,k_a,k-k_a,0)$:

$$h_{ABC}(n,m,l,k_a,k-k_a,0)=h_{AB}(n,m,l,k_a,k-k_a).$$

Applying Equation (3) for $h_{AB}(n,m,l,k_a,k-k_a)$, we obtain

$$h_{AB}(n,m,l,k_a,k-k_a)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{A}^{\Delta }(n_i,m_i,l_i,k_a)·B^{\Delta }(n-n_i,m-m_i,l-l_i,k-k_a).$$

Combining the obtained formulas, we derive the desired result that is presented in Equation (5). □

In a similar way, we can prove the rules for calculating the coefficients of the generating function that is the result of the addition of two generating functions in three variables.

2.3. Multiplication of Generating Functions in Three Variables

Next, let us consider the process of obtaining an explicit formula for the composita of the generating function in three variables that is the result of the multiplication of two generating functions in three variables.

Theorem 4. 

Suppose that:

$$A{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{A}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(A\right)\ge 1,$$

$$B{(x,y,z)}^k=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}{B}^{\Delta }(n,m,l,k)x^n{y}^m{z}^l,ord\left(B\right)\ge 1.$$

Then, the composita $G^{\Delta }(n,m,l,k)$ of the multiplication of generating functions in three variables

$$G(x,y,z)=A(x,y,z)·B(x,y,z)=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}g(n,m,l)x^n{y}^m{z}^l$$

is equal to

$$G^{\Delta }(n,m,l,k)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{A}^{\Delta }(n_i,m_i,l_i,k)·B^{\Delta }(n-n_i,m-m_i,l-l_i,k).$$

(6)

Proof. 

To obtain the composita $G^{\Delta }(n,m,l,k)$, we can represent the generating function $G(x,y,z)$ as the following composition of generating functions:

$$G(x,y,z)=A(x,y,z)·B(x,y,z)=H\left(A\right(x,y,z),B(x,y,z),C(x,y,z\left)\right),$$

where

$$H(x,y,z)=xy{z}^0=xy,$$

$$H^{\Delta }(n,m,l,k)=\delta (n,k)\delta (m,k)\delta (l,0).$$

Applying Theorem 2 for this composition, we derive the desired result presented in Equation (6).

Moreover, we can obtain the same result by applying the convolution operation to

$$G{(x,y,z)}^k=A{(x,y,z)}^k·B{(x,y,z)}^k,$$

and we also obtain Equation (6). □

2.4. Recurrence for Calculating Compositae of Generating Functions in Three Variables

Generating functions are widely used when working with recurrence relations. It is also convenient to use recurrences when studying changes in combinatorial objects caused by a change in their parameters. Therefore, there is a need to obtain such recurrences for the coefficients of generating functions.

If we represent the k-th power of $F(x,y,z)$ as

$$F{(x,y,z)}^k=F(x,y,z)F{(x,y,z)}^{k-1},F{(x,y,z)}^0=1.$$

and apply the convolution operation, then we obtain the following recurrence for calculating compositae:

$$F^{\Delta }(n,m,l,k)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^{l}f(n_i,n_j,l_i)F^{\Delta }(n-n_i,m-m_i,l-l_i,k-1),$$

$$F^{\Delta }(n,m,l,1)=f(n,m,l).$$

Hence, if we have an explicit formula or recurrence for the coefficients of a generating function in three variables, then we can also obtain a recurrence for the composita of this generating function. Moreover, such a recurrence can later be applied to obtain a recurrence for the coefficients of the generating function in the calculation of which this composita is used.

3. Method for Obtaining Coefficients of Rational Generating Functions in $\mathit{n}$ Variables

Thus, if a given generating function in three variables can be represented as a composition of simpler generating functions with known formulas for their coefficients or compositae, then we can obtain an explicit formula or recurrence for the coefficients or composita of this generating function.

In a similar way, operations on compositae can be expanded to the case of generating functions in n variables. Let us consider a general scheme for obtaining the coefficients of a rational generating function in n variables.

A rational generating function in n variables is the following formal power series [26]:

$$F(x_1,x_2,\dots ,x_n)=\sum _{k_1\ge 0}\sum _{k_2\ge 0}\dots \sum _{k_n\ge 0}f(k_1,k_2,\dots ,k_n)x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n}=\frac{P(x_1,x_2,\dots ,x_n)}{Q(x_1,x_2,\dots ,x_n)},$$

where $P(x_1,x_2,\dots ,x_n)$ and $Q(x_1,x_2,\dots ,x_n)$ are generating functions in n variables

$$P(x_1,x_2,\dots ,x_n)=\sum _{k_1\ge 0}\sum _{k_2\ge 0}\dots \sum _{k_n\ge 0}p(k_1,k_2,\dots ,k_n)x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n},$$

$$Q(x_1,x_2,\dots ,x_n)=\sum _{k_1\ge 0}\sum _{k_2\ge 0}\dots \sum _{k_n\ge 0}q(k_1,k_2,\dots ,k_n)x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n}.$$

Then, to obtain an explicit formula for the coefficients $f(k_1,k_2,\dots ,k_n)$, we can use the following method:

• Represent the generating function $P(x_1,x_2,\dots ,x_n)$ as the following way of addition of generating functions $P_i(x_1,x_2,\dots ,x_n)$, where the coefficients $p_i(k_1,k_2,\dots ,k_n)$ and compositae $P_i^{\Delta }(k_1,k_2,\dots ,k_n)$ can be easily obtained:

  $$P(x_1,x_2,\dots ,x_n)=\sum _i{P}_i(x_1,x_2,\dots ,x_n);$$
• Obtain explicit formulas or recurrences for the coefficients $p_i(k_1,k_2,\dots ,k_n)$ and compositae $P_i^{\Delta }(k_1,k_2,\dots ,k_n)$;
• Represent the generating function $Q(x_1,x_2,\dots ,x_n)$ as the following way of addition of generating functions $Q_i(x_1,x_2,\dots ,x_n)$, where the coefficients $q_i(k_1,k_2,\dots ,k_n)$ and compositae $Q_i^{\Delta }(k_1,k_2,\dots ,k_n)$ can be easily obtained:

  $$Q(x_1,x_2,\dots ,x_n)=1-\sum _i{Q}_i(x_1,x_2,\dots ,x_n);$$
• Obtain explicit formulas or recurrences for the coefficients $q_i(k_1,k_2,\dots ,k_n)$ and compositae $Q_i^{\Delta }(k_1,k_2,\dots ,k_n)$;
• Sequentially performing composition and addition operations for each generating function $Q_i(x_1,x_2,\dots ,x_n)$, obtain an explicit formula or recurrence for the composita $Q_s^{\Delta }(k_1,k_2,\dots ,k_n,k)$ of

  $$Q_s(x_1,x_2,\dots ,x_n)=\sum _i{Q}_i(x_1,x_2,\dots ,x_n);$$
• Calculate the coefficients of

  $$R(x_1,x_2,\dots ,x_n)=\sum _{k_1\ge 0}\sum _{k_2\ge 0}\dots \sum _{k_n\ge 0}r(k_1,k_2,\dots ,k_n)x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n}=$$

  $$=\frac{1}{Q(x_1,x_2,\dots ,x_n)}=\frac{1}{1-\sum _i{Q}_i(x_1,x_2,\dots ,x_n)}=\frac{1}{1-Q_s(x_1,x_2,\dots ,x_n)}$$

  by applying

  $$r(k_1,k_2,\dots ,k_n)=\sum _{k=0}^{k_1+k_2+\dots +k_n}{Q}_s^{\Delta }(k_1,k_2,\dots ,k_n,k);$$

  (7)
• Calculate the coefficients of

  $$F(x_1,x_2,\dots ,x_n)=\frac{P(x_1,x_2,\dots ,x_n)}{Q(x_1,x_2,\dots ,x_n)}=P(x_1,x_2,\dots ,x_n)·R(x_1,x_2,\dots ,x_n)$$

  by applying the convolution operation, that is,

  $$f(k_1,k_2,\dots ,k_n)=\sum _{l_1\ge 0}\sum _{l_2\ge 0}\dots \sum _{l_n\ge 0}p(l_1,l_2,\dots ,l_n)r(k_1-l_1,k_2-l_2,\dots ,k_n-l_n).$$

As a result of performing all the required actions, an explicit formula or recurrence for the coefficients of a given rational generating function in n variables will be obtained. Note that the main task is to decompose the original expression of a given multivariate generating function into simpler ones.

4. Application of Compositae for Obtaining Coefficients of Generating Functions in Three Variables

Next, let us consider examples of applying the proposed method for obtaining coefficients of multivariate generating functions.

4.1. Example 1

Let us consider the following generating function in three variables:

$$G(x,y,z)=\frac{\frac{z}{2}}{(1-yz)(1-(x+\frac{1}{x}+y+\frac{1}{y})\frac{z}{2}+z^2)}=\sum _{n>0}\sum _{i=-n}^n\sum _{j=-n}^n\rho (i,j,n)x^i{y}^j{z}^n,$$

where $\rho (i,j,n)$ is the north-going edge probability for the cell centered at $(i,j)$ in an Aztec diamond of order n [27].

To remove negative powers of variables, consider the following generating function:

$$F(x,y,z)=G(x,y,xyz)=\sum _{n>0}\sum _{i=-n}^n\sum _{j=-n}^n\rho (i,j,n)x^i{y}^j{\left(xyz\right)}^n=$$

$$=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l>0}\rho (n-l,m-l,l)x^n{y}^m{z}^l=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l>0}f(n,m,l)x^n{y}^m{z}^l.$$

According to the proposed method for obtaining coefficients of multivariate generating functions, we represent the generating function $F(x,y,z)$ as a rational generating function in three variables

$$F(x,y,z)=\frac{\frac{xyz}{2}}{(1-x{y}^{2}z)(1-(x+\frac{1}{x}+y+\frac{1}{y})\frac{xyz}{2}+x^2{y}^2{z}^2)}=\frac{P(x,y,z)}{Q(x,y,z)},$$

where we use the following decomposition of generating functions:

$$P(x,y,z)=\frac{\frac{xyz}{2}}{1-x{y}^{2}z}=\frac{1}{2}{P}_1(x,y,z)P_2(x,y,z)=\frac{1}{2}{P}_1(x,y,z)\frac{1}{1-P_3(x,y,z)},$$

$$P_1(x,y,z)=xyz,P_2(x,y,z)=\frac{1}{1-x{y}^{2}z},P_3(x,y,z)=x{y}^{2}z,$$

$$Q(x,y,z)=1-\left(x+\frac{1}{x}+y+\frac{1}{y}\right)\frac{xyz}{2}+x^2{y}^2{z}^2=1-(Q_1(x,y,z)+Q_2(x,y,z)),$$

$$Q_1(x,y,z)=\frac{1}{2}(x+y)(1+xy)z=\frac{1}{2}{P}_1(x+y,1+xy,z),$$

$$Q_2(x,y,z)=-x^2{y}^2{z}^2=-P_1{(x,y,z)}^2.$$

Next, we obtain the required coefficients for generating functions that are used in the presented decomposition.

For the generating function $P_1(x,y,z)$, its coefficients are

$$p_1(n,m,l)=\delta (n,1)\delta (m,1)\delta (l,1)$$

and its composita is

$$P_1^{\Delta }(n,m,l,k)=\delta (n,k)\delta (m,k)\delta (l,k).$$

For the generating function $P_3(x,y,z)$, its coefficients are

$$p_3(n,m,l)=\delta (n,1)\delta (m,2)\delta (l,1)$$

and its composita is

$$P_3^{\Delta }(n,m,l,k)=\delta (n,k)\delta (m,2k)\delta (l,k).$$

According to Equation (7), the coefficients of the generating function $P_2(x,y,z)$ are

$$p_2(n,m,l)=\sum _{k=0}^{n+m+l}{P}_3^{\Delta }(n,m,l,k)=\sum _{k=0}^{n+m+l}\delta (n,k)\delta (m,2k)\delta (l,k)=\delta (m,2n)\delta (l,n).$$

Thus, applying the convolution operation, we obtain the following formula for the coefficients of the generating function $P(x,y,z)$:

$$p(n,m,l)=\frac{1}{2}\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{p}_1(n_i,m_i,l_i)·p_2(n-n_i,m-m_i,l-l_i)=$$

$$=\frac{1}{2}\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l\delta (n_i,1)\delta (m_i,1)\delta (l_i,1)\delta (m-m_i,2n-2n_i)\delta (l-l_i,n-n_i)=$$

$$=\frac{1}{2}\delta (m,2n-1)\delta (l,n).$$

To obtain the composita of the generating function $Q_1(x,y,z)$, we use Theorem 2, where

$$H(x,y,z)=\frac{1}{2}xyz=\frac{1}{2}{P}_1(x,y,z),H^{\Delta }(n,m,l,k)=\frac{1}{2^k}{P}_1^{\Delta }(n,m,l,k),$$

$$A(x,y,z)=x+y,A^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{k}{n}\right)\delta (m,k-n)\delta (l,0),$$

$$B(x,y,z)=1+xy,B^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{k}{n}\right)\delta (m,n)\delta (l,0),$$

$$C(x,y,z)=z,C^{\Delta }(n,m,l,k)=\delta (n,0)\delta (m,0)\delta (l,k).$$

Then, we find the components required for Theorem 2:

$$h_{AB}(n,m,l,k_a,k_b)=$$

$$=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l\left(\genfrac{}{}{0pt}{}{k_a}{n_i}\right)\delta (m_i,k_a-n_i)\delta (l_i,0)\left(\genfrac{}{}{0pt}{}{k_b}{n-n_i}\right)\delta (m-m_i,n-n_i)\delta (l-l_i,0)=$$

$$=\sum _{n_i=0}^n\left(\genfrac{}{}{0pt}{}{k_a}{n_i}\right)\left(\genfrac{}{}{0pt}{}{k_b}{n-n_i}\right)\delta (2n_i,n-m+k_a)\delta (l,0)=$$

$$=\left(\genfrac{}{}{0pt}{}{k_a}{\frac{n-m+k_a}{2}}\right)\left(\genfrac{}{}{0pt}{}{k_b}{\frac{n+m-k_a}{2}}\right)\frac{{(-1)}^{n-m+k_a}+1}{2}\delta (l,0).$$

$$h_{ABC}(n,m,l,k_a,k_b,k_c)=$$

$$=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{h}_{AB}(n_i,m_i,l_i,k_a,k_b)\delta (n-n_i,0)\delta (m-m_i,0)\delta (l-l_i,k_c)=$$

$$=h_{AB}(n,m,l-k_c,k_a,k_b).$$

Thus, we obtain

$$Q_1^{\Delta }(n,m,l,k)=$$

$$=\frac{1}{2^k}\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}{P}_1^{\Delta }(k_a,k_b,k_c,k)·h_{ABC}(n,m,l,k_a,k_b,k_c)=$$

$$=\frac{1}{2^k}\sum _{k_a=0}^{n+m+l}\sum _{k_b=0}^{n+m+l-k_a}\sum _{k_c=0}^{n+m+l-k_a-k_b}\delta (k_a,k)\delta (k_b,k)\delta (k_c,k)h_{AB}(n,m,l-k_c,k_a,k_b)=$$

$$=\frac{1}{2^k}{h}_{AB}(n,m,l-k,k,k)=\left(\genfrac{}{}{0pt}{}{k}{\frac{n-m+k}{2}}\right)\left(\genfrac{}{}{0pt}{}{k}{\frac{n+m-k}{2}}\right)\frac{{(-1)}^{n-m+k}+1}{2^{k+1}}\delta (l,k).$$

For the generating function $Q_2(x,y,z)$, its composita is

$$Q_2^{\Delta }(n,m,l,k)={(-1)}^k\delta (n,2k)\delta (m,2k)\delta (l,2k).$$

For the addition of the generating functions $Q_s(x,y,z)=Q_1(x,y,z)+Q_2(x,y,z)$, we use Theorem 3 and obtain its composita:

$$Q_s^{\Delta }(n,m,l,k)=\sum _{k_a=0}^{n+m+l}\left(\genfrac{}{}{0pt}{}{k}{k_a}\right)\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{Q}_1^{\Delta }(n_i,m_i,l_i,k_a)·Q_2^{\Delta }(n-n_i,m-m_i,l-l_i,k-k_a).$$

After simplifying, we have

$$Q_s^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{k}{l-k}\right)\left(\genfrac{}{}{0pt}{}{2k-l}{\frac{n-m-l+2k}{2}}\right)\left(\genfrac{}{}{0pt}{}{2k-l}{\frac{n+m-3l+2k}{2}}\right)\frac{{(-1)}^{n+m+k}+{(-1)}^{l+k}}{2^{2k-l+1}}.$$

According to Equation (7), the coefficients of the generating function

$$R(x,y,z)=\frac{1}{Q(x,y,z)}=\frac{1}{1-(Q_1(x,y,z)+Q_2(x,y,z))}=\frac{1}{1-Q_s(x,y,z)}$$

are

$$r(n,m,l)=\sum _{k=0}^{n+m+l}{Q}_s^{\Delta }(n,m,l,k)=$$

$$=\sum _{k=0}^{n+m+l}\left(\genfrac{}{}{0pt}{}{k}{l-k}\right)\left(\genfrac{}{}{0pt}{}{2k-l}{\frac{n-m-l+2k}{2}}\right)\left(\genfrac{}{}{0pt}{}{2k-l}{\frac{n+m-3l+2k}{2}}\right)\frac{{(-1)}^{n+m+k}+{(-1)}^{l+k}}{2^{2k-l+1}}.$$

Finally, applying the convolution operation, we obtain the following formula for the coefficients of the generating function $F(x,y,z)$:

$$f(n,m,l)=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^{l}p(n_i,m_i,l_i)·r(n-n_i,m-m_i,l-l_i)=$$

$$=\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l\frac{1}{2}\delta (m_i,2n_i-1)\delta (l_i,n_i)·r(n-n_i,m-m_i,l-l_i)=$$

$$=\frac{1}{2}\sum _{n_i=1}^{n}r(n-n_i,m-2n_i+1,l-n_i).$$

Combining all the obtained results, we can calculate the north-going edge probability for the cell centered at $(i,j)$ in an Aztec diamond of order n by using the following formula:

$$\rho (i,j,n)=f(n+i,n+j,n).$$

4.2. Example 2

Let us consider the following generating function in three variables:

$$G(x,y,z)=\frac{exp\left((x-1)z+(y-1)\frac{z^2}{2}\right)}{1-z}=\sum _{n\ge 0}\sum _{m\ge 0}\sum _{l\ge 0}\frac{g(n,m,l)}{l!}{x}^n{y}^m{z}^l,$$

where $g(n,m,l)$ is equal to the number of permutations of l elements with n 1-cycles and m 2-cycles [[28] Note III.5].

According to the proposed method for obtaining coefficients of multivariate generating functions, we represent the generating function $F(x,y,z)$ as a rational generating function in three variables

$$G(x,y,z)=\frac{exp\left((x-1)z+(y-1)\frac{z^2}{2}\right)}{1-z}=P(x,y,z)·R(x,y,z),$$

where we use the following decomposition of generating functions:

$$P(x,y,z)=exp\left((x-1)z+(y-1)\frac{z^2}{2}\right)=P_1\left(P_2(x,y,z)\right),$$

$$P_1\left(x\right)=e^x,P_2(x,y,z)=(x-1)z+(y-1)\frac{z^2}{2}=P_3(x,y,z)+P_4(x,y,z),$$

$$P_3(x,y,z)=(x-1)y^{0}z=(x-1)z,P_4(x,y,z)=x^0(y-1)\frac{z^2}{2}=(y-1)\frac{z^2}{2},$$

$$R(x,y,z)=\frac{x^0{y}^0}{1-z}=\frac{1}{1-z}.$$

Next, we obtain the required coefficients for generating functions that are used in the presented decomposition.

Based on the binomial theorem, the generating functions $P_3(x,y,z)$ and $P_4(x,y,z)$ have the following compositae:

$$P_3^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{k}{n}\right){(-1)}^{n-k}\delta (m,0)\delta (l,k),$$

$$P_4^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{k}{m}\right)\frac{{(-1)}^{m-k}}{2^k}\delta (n,0)\delta (l,2k).$$

For the addition of the generating functions $P_2(x,y,z)=P_3(x,y,z)+P_4(x,y,z)$, we use Theorem 3 and obtain its composita

$$P_2^{\Delta }(n,m,l,k)=\sum _{k_a=0}^{n+m+l}\left(\genfrac{}{}{0pt}{}{k}{k_a}\right)\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^l{P}_3^{\Delta }(n_i,m_i,l_i,k_a)·P_4^{\Delta }(n-n_i,m-m_i,l-l_i,k-k_a).$$

After simplifying, we have

$$P_2^{\Delta }(n,m,l,k)=\left(\genfrac{}{}{0pt}{}{2k-l}{n}\right)\left(\genfrac{}{}{0pt}{}{l-k}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{l-k}\right)\frac{{(-1)}^{n+m+k}}{2^{l-k}}.$$

For the generating function $P_1\left(x\right)$, its coefficients are

$$p_1\left(n\right)=\frac{1}{n!}.$$

For the composition of the generating functions $P(x,y,z)=P_1\left(P_2(x,y,z)\right)$, we use Theorem 1 and obtain its coefficients

$$p(n,m,l)=\sum _{k=0}^{n+m+l}{p}_1\left(k\right)·P_2^{\Delta }(n,m,l,k)=\sum _{k=0}^{l-m}\left(\genfrac{}{}{0pt}{}{2k-l}{n}\right)\left(\genfrac{}{}{0pt}{}{l-k}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{l-k}\right)\frac{{(-1)}^{n+m+k}}{k!2^{l-k}}.$$

For the generating function $R(x,y,z)$, its coefficients are

$$r(n,m,l)=\delta (n,0)\delta (m,0).$$

Finally, applying the convolution operation, we obtain the following formula for the coefficients of the generating function $G(x,y,z)$:

$$g(n,m,l)=l!\sum _{n_i=0}^n\sum _{m_i=0}^m\sum _{l_i=0}^{l}p(n_i,m_i,l_i)·r(n-n_i,m-m_i,l-l_i)=$$

$$=l!\sum _{l_i=0}^{l}p(n,m,l_i)=l!\sum _{l_i=0}^l\sum _{k=0}^{l_i-m}\left(\genfrac{}{}{0pt}{}{2k-l_i}{n}\right)\left(\genfrac{}{}{0pt}{}{l_i-k}{m}\right)\left(\genfrac{}{}{0pt}{}{k}{l_i-k}\right)\frac{{(-1)}^{n+m+k}}{k!2^{l_i-k}}.$$

5. Discussion

There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions by using the powers of generating functions in one variable. One such concept is the application of compositae of generating functions. The operations on the compositae of generating functions allow for obtaining explicit formulas and recurrences for coefficients and compositae of generating functions. Our previous studies have shown the applicability of compositae for the case of bivariate generating functions (i.e., for generating functions in two variables). In this paper, we have expanded these operations on compositae to the case of multivariate generating functions. To do this, we have considered in detail the case of generating functions in three variables and then presented a general method for obtaining coefficients of rational generating functions in n variables.

The main restriction for applying the proposed method is: if a given multivariate generating function can be represented as a composition of simpler generating functions with known formulas for their coefficients or compositae, then an explicit formula or recurrence for the coefficients or composita of this generating function can be obtained. The mathematical apparatus of compositae of multivariate generating functions can be applied in different mathematical problems where generating functions are used. For example, if we have a multivariate generating function of the sequence of integers, each of which is equal to the number of objects in a given combinatorial set with fixed parameters, then we can obtain new expressions for the cardinality function of this combinatorial set. If the coefficients of a multivariate generating function are associated with some other mathematical structure, then the proposed method allows obtaining explicit formulas or recurrences for calculating such structures.

To confirm the effectiveness of applying the proposed method, we also present examples of obtaining new explicit formulas for the coefficients of multivariate generating functions. In Example 1, we find a new explicit formula for the coefficients of a generating function connected with an Aztec diamond. In Example 2, we find a new explicit formula for the coefficients of a generating function connected with the cardinality function of the combinatorial set of permutations with cycles.

Thus, the main contribution of this article is the proposed method for obtaining coefficients of rational generating functions in n variables. The essential feature of this method is its generality and the possibility of application to multivariate generating functions. The scheme used in this article to obtain explicit formulas for the coefficients of generating functions in three variables can also be applied to the generating functions in n variables where $n>3$.