Unexpected Biases between Congruence Classes for Parts in k-indivisible Partitions
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| Format: | Preprint |
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2022
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| _version_ | 1866908563986710528 |
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| author | Jackson, Faye Otgonbayar, Misheel |
| author_facet | Jackson, Faye Otgonbayar, Misheel |
| contents | 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 $k \geq \frac{6(t^2-1)}{π^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k < \frac{6(t^2-1)}{π^2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_06365 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unexpected Biases between Congruence Classes for Parts in k-indivisible Partitions Jackson, Faye Otgonbayar, Misheel Combinatorics Number Theory Primary: 11P82. Secondary: 05A17, 11P81 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 $k \geq \frac{6(t^2-1)}{π^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k < \frac{6(t^2-1)}{π^2}$. |
| title | Unexpected Biases between Congruence Classes for Parts in k-indivisible Partitions |
| topic | Combinatorics Number Theory Primary: 11P82. Secondary: 05A17, 11P81 |
| url | https://arxiv.org/abs/2207.06365 |