Method for Obtaining Coefficients of Powers of Multivariate Generating Functions
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 co...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-06-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/13/2859 |
| _version_ | 1797591298576744448 |
|---|---|
| author | Dmitry Kruchinin Vladimir Kruchinin Yuriy Shablya |
| author_facet | Dmitry Kruchinin Vladimir Kruchinin Yuriy Shablya |
| author_sort | Dmitry Kruchinin |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T01:35:26Z |
| format | Article |
| id | doaj.art-a529c2256b614a1ca4ef5947ef55bfea |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T01:35:26Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a529c2256b614a1ca4ef5947ef55bfea2023-11-18T17:02:23ZengMDPI AGMathematics2227-73902023-06-011113285910.3390/math11132859Method for Obtaining Coefficients of Powers of Multivariate Generating FunctionsDmitry Kruchinin0Vladimir Kruchinin1Yuriy Shablya2Laboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, RussiaLaboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, RussiaLaboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, RussiaThere 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.https://www.mdpi.com/2227-7390/11/13/2859multivariate generating functioncoefficientpower of generating functioncomposition of generation functionscompositaexplicit formula |
| spellingShingle | Dmitry Kruchinin Vladimir Kruchinin Yuriy Shablya Method for Obtaining Coefficients of Powers of Multivariate Generating Functions Mathematics multivariate generating function coefficient power of generating function composition of generation functions composita explicit formula |
| title | Method for Obtaining Coefficients of Powers of Multivariate Generating Functions |
| title_full | Method for Obtaining Coefficients of Powers of Multivariate Generating Functions |
| title_fullStr | Method for Obtaining Coefficients of Powers of Multivariate Generating Functions |
| title_full_unstemmed | Method for Obtaining Coefficients of Powers of Multivariate Generating Functions |
| title_short | Method for Obtaining Coefficients of Powers of Multivariate Generating Functions |
| title_sort | method for obtaining coefficients of powers of multivariate generating functions |
| topic | multivariate generating function coefficient power of generating function composition of generation functions composita explicit formula |
| url | https://www.mdpi.com/2227-7390/11/13/2859 |
| work_keys_str_mv | AT dmitrykruchinin methodforobtainingcoefficientsofpowersofmultivariategeneratingfunctions AT vladimirkruchinin methodforobtainingcoefficientsofpowersofmultivariategeneratingfunctions AT yuriyshablya methodforobtainingcoefficientsofpowersofmultivariategeneratingfunctions |