论文标题
k型分区中零件的一致班级之间的偏见
Biases among Congruence Classes for Parts in k-regular Partitions
论文作者
论文摘要
对于整数$ k，t \ geq 2 $和$ 1 \ leq r \ leq r \ leq t $ let $ d_k（r，t; n）$是所有$ k $ regular-regular-regular-regular-grounder comptitions中的零件数（即所有零件的分区，所有零件的分区小于$ k $ by $ k $ y BULS $ n $），这是$ n $的$ n $ y modul $ r $ r $ r $ r $ r $ r $ r $ t $。使用圆法，我们获得了渐近学 \ [ d_ {k}（r，t; n）= \ frac {3^{\ frac {\ frac {1} {4}} e^{π\ sqrt {\ frac {\ frac {2kn} {3}}}}}}}}}}}}}}}}}} {πt 2^{\ frac {3} {4}} k^{\ frac {\ frac {1} {4}} n^{\ frac {\ frac {1} {4}} {4}} \ sqrt {k}}} \ left（ k} {8 \ sqrt {6}π} - \ frac {tπ（k-1）k^{\ frac {\ frac {1} {2}}}}} {2 \ sqrt {6}}}}} \ left（ \ frac {1} {2} \ right）\ right）n^{ - \ frac {1} {2}}}}} + o（n^{ - 1}）\ right）， \]其中$ k = 1- \ frac {1} {k} $。该渐近性的主要术语不取决于$ r $，因此，如果$ p_k（n）$是$ n $ $ n $的所有$ k $ regartular分区中的零件总数，我们有$ \ frac {d_k（r，t; n）} {p_k（n）} {p_k（n）}} \ to \ frac \ frac \ frac \ frac {t} $} $。因此，从薄弱的渐近意义上讲，这些部分在一致性类别之间被等待。但是，对较低阶段的检查表明对较低的一致性类别有偏见。也就是说，对于$ 1 \ leq r <s \ leq t $，我们有$ d_k（r，t; n）\ geq d_k（s，t; n）$，用于足够大的$ n $。我们明确说明了这种不平等，表明$ 3 \ leq k \ leq 10 $和$ 2 \ leq t \ leq t \ leq 10 $不等式$ d_k（r，t; n）\ geq d_k（s，t; n）$持有所有$ n \ geq 1 $ and geq 1 $ and the Strict n \ d_k $ d_k（r，t $ nk）$ nk（r，t; \ geq 17 $。
For integers $k,t \geq 2$ and $1\leq r \leq t$ let $D_k(r,t;n)$ be the number of parts among all $k$-regular partitions (i.e., partitions of $n$ where all parts have multiplicity less than $k$) of $n$ that are congruent to $r$ modulo $t$. Using the circle method, we obtain the asymptotic \[ D_{k}(r,t;n) = \frac{3^{\frac{1}{4}}e^{π\sqrt{\frac{2Kn}{3}}}}{πt 2^{\frac{3}{4}}K^{\frac{1}{4}}n^{\frac{1}{4}}\sqrt{k}}\left(\log k + \left(\frac{3\sqrt{K}\log k}{8\sqrt{6}π} - \frac{tπ(k-1)K^{\frac{1}{2}}}{2\sqrt{6}}\left(\frac{r}{t}- \frac{1}{2}\right)\right)n^{-\frac{1}{2}} + O(n^{-1})\right), \] where $K = 1 - \frac{1}{k}$. The main term of this asymptotic does not depend on $r$, and so if $P_k(n)$ is the total number of parts among all $k$-regular partitions of $n$, we have that $\frac{D_k(r,t;n)}{P_k(n)} \to \frac{1}{t}$ as $n \to \infty$. Thus, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards the lower congruence classes; that is, for $1\leq r < s \leq t$ we have $D_k(r,t;n) \geq D_k(s,t;n)$ for sufficiently large $n$. We make this inequality explicit, showing that for $3 \leq k \leq 10$ and $2 \leq t \leq 10$ the inequality $D_k(r,t;n) \geq D_k(s,t;n)$ holds for all $n \geq 1$ and the strict inequality $D_k(r,t;n) > D_k(s,t;n)$ holds for all $n \geq 17$.
