论文标题
k难度分区中零件的一致性课程之间的意外偏见
Unexpected Biases between Congruence Classes for Parts in k-indivisible Partitions
论文作者
论文摘要
对于整数，$ k，t \ geq 2 $，$ 1 \ leq r \ leq t $ let $ d_k^\ times（r，t; n）$是所有$ k $ n $ $ n $的所有$ k $ n $中的零件数（即，所有零件都不划分为$ n $ $ n $ $ n $ $ n $ $ r $ r r t $ n $ n $ n $ n $ n $ n $ n $ r r ry $ r r r r ry。使用Wright的Circle方法，我们得出了$ d_k^\ times（r，t; n）$作为$ n \ to \ infty $的渐近造成的，当$ k，t $是coprime。该渐近性的主要术语不取决于$ r $，因此，从薄渐近的意义上讲，这些零件在一致性类别之间被等待。但是，对较低顺序条款的检查表明，对不同的一致性类模型$ t $有偏见。这会引起一致性类Modulo $ t $的订购，我们称之为$ k $ - 不可分割的订购。我们证明，对于$ k \ geq \ frac {6（t^2-1）}} {π^2} $ $ k $ - 不细分的订购与自然订购相匹配。当$ k <\ frac {6（t^2-1）} {π^2} $时，我们还探讨了这些顺序的属性。
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 $k < \frac{6(t^2-1)}{π^2}$.
