The $n$-Color Partition Function and Some Counting Theorems
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909303430971392 |
|---|---|
| author | Bandyopadhyay, Subhajit Baruah, Nayandeep Deka |
| author_facet | Bandyopadhyay, Subhajit Baruah, Nayandeep Deka |
| contents | Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical Möbius function as well as Euler's totient. In this paper, we define a counting function $T_k^r(m)$ on the set of $n$-color partitions of $m$ for given positive integers $k, r$ and relate the function with the $n$-color partition function and other well-known arithmetic functions like the Möbius function, Liouville function, etc. and their divisor sums. Furthermore, we use a counting method of Erdös to obtain some counting theorems for $n$-color partitions that are analogous to those found by Andrews and Deutsch for the partition function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02004 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $n$-Color Partition Function and Some Counting Theorems Bandyopadhyay, Subhajit Baruah, Nayandeep Deka Combinatorics Number Theory 05A17 (Primary) 11P81, 11P83, 11B37 (Secondary) Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical Möbius function as well as Euler's totient. In this paper, we define a counting function $T_k^r(m)$ on the set of $n$-color partitions of $m$ for given positive integers $k, r$ and relate the function with the $n$-color partition function and other well-known arithmetic functions like the Möbius function, Liouville function, etc. and their divisor sums. Furthermore, we use a counting method of Erdös to obtain some counting theorems for $n$-color partitions that are analogous to those found by Andrews and Deutsch for the partition function. |
| title | The $n$-Color Partition Function and Some Counting Theorems |
| topic | Combinatorics Number Theory 05A17 (Primary) 11P81, 11P83, 11B37 (Secondary) |
| url | https://arxiv.org/abs/2409.02004 |