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Unexpected biases between congruence classes for parts in k-indivisible partitions
- Source :
- Journal of Number Theory. 248:310-342
- Publication Year :
- 2023
- Publisher :
- Elsevier BV, 2023.
-
Abstract
- For integers $k,t \geq 2$, and $1\leq r \leq t$ let $D_k^\times(r,t;n)$ be the number of parts among all $k$-indivisible partitions of $n$ (i.e., partitions where all parts are not divisible by $k$) of $n$ that are congruent to $r$ modulo $t$. Using Wright's circle method, we derive an asymptotic for $D_k^\times(r,t;n)$ as $n \to \infty$ when $k,t$ are coprime. The main term of this asymptotic does not depend on $r$, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo $t$. This induces an ordering on the congruence classes modulo $t$, which we call the $k$-indivisible ordering. We prove that for $k \geq \frac{6(t^2-1)}{\pi^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k < \frac{6(t^2-1)}{\pi^2}$.<br />Comment: 26 pages, 3 figures
Details
- ISSN :
- 0022314X
- Volume :
- 248
- Database :
- OpenAIRE
- Journal :
- Journal of Number Theory
- Accession number :
- edsair.doi.dedup.....ade901fe12c950aeeaaac668c195aea4