Partitions into a small number of part sizes
| Autor: | William J. Keith |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Generality Algebra and Number Theory Small number 010102 general mathematics Modular form 0102 computer and information sciences Function (mathematics) Congruence relation 01 natural sciences 05A17 11P83 010201 computation theory & mathematics FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Mathematics |
| Zdroj: | International Journal of Number Theory. 13:229-241 |
| ISSN: | 1793-7310 1793-0421 |
| DOI: | 10.1142/s1793042117500130 |
| Popis: | We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0 \pmod{4}$ for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many other progressions for which our method should easily generalize. Of some independent interest, we prove that the overpartition function $\bar{p}(n) \equiv 0 \pmod{16}$ in the first three progressions (the fourth is known), and thereby show that $\nu_3(An+B) \equiv 0 \pmod{2}$ in each of these progressions as well, and discuss the relationship between these congruences in more generality. We end with open questions in this area. Comment: 11 pages; v2, small correction to proof of Theorem 7; v3, clean up some explanations, acknowledge recent results from Xinhua Xiong on overpartitions mod 16; v4, final journal version to appear International Journal of Number Theory (Feb. 2017) |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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