论文标题
关于康和公园的猜想的概括
On Generalizations of a Conjecture of Kang and Park
论文作者
论文摘要
令$δ_d^{（a， - ）}（n）= q_d^{（a）}（n） - q_d^{（a， - ）}（n）$，其中$ q_d^{（a）}（a）}（a）}（n）$计数$ n $的零件数量至少和$ d $ d $ d $ d $ n $ n $ n $ a n $ a n $ a n $ a n $ a n $ a n $ a n $ a n和q a n $ q.将分区数计为$ \ equiv \ pm a \ pmod {d + 3} $不包括$ d + 3-a $ part。通过概括了康和帕克，邓肯，昆格，史威尔的猜想，第二位作者推测$δ_d^{（3， - - - ）}（3， - ）}（n）\ geq 0 $ for $ d \ geq 1 $ and $ n \ geq 1 $ and $ n \ geq 1 $，并且可以在$ d $ d $ dives $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3 $ 3上prove。 They were also able to conjecture an analog for higher values of $a$ that the modified difference function $Δ_{d}^{(a,-,-)}(n) = q_{d}^{(a)}(n) - Q_{d}^{(a,-,-)}(n) \geq 0$ where $ q_ {d}^{（a， - ， - ， - ）}（n）$将分区数计为零件$ \ equiv \ equiv \ pm a \ pmod {d + 3} $不包括$ a $ a $ a $ a $ a和$ d + 3-a $零件，并为无限多个类别的$ n $和$ d $提供了它。我们证明$δ_{d}^{（3， - ）}（n）\ geq 0 $，除了有限的许多$ d $。我们还为所有人提供了广义猜想的证明，除了有限的$ d $，固定$ a $，并加强了邓肯，昆格，斯威舍和第二作者的结果。我们为广义猜想提供了$ D $线性下限的有条件证明，这改善了我们的无条件结果，基于对alder的最近经过验证的猜想的猜想。使用这种修改，我们可以加强对Kang和Park的猜想的这种概括，这非常允许$ a $作为一部分。此外，我们提供了渐近证据表明这种增强的猜想所拥有的。
Let $Δ_d^{(a,-)}(n) = q_d^{(a)}(n) - Q_d^{(a,-)}(n)$ where $q_d^{(a)}(n)$ counts the number of partitions of $n$ into parts with difference at least $d$ and size at least $a$, and $Q_d^{(a,-)}(n)$ counts the number of partitions into parts $\equiv \pm a \pmod{d + 3}$ excluding the $d+3-a$ part. Motivated by generalizing a conjecture of Kang and Park, Duncan, Khunger, Swisher, and the second author conjectured that $Δ_d^{(3,-)}(n)\geq 0$ for all $d\geq 1$ and $n\geq 1$ and were able to prove this when $d \geq 31$ is divisible by $3$. They were also able to conjecture an analog for higher values of $a$ that the modified difference function $Δ_{d}^{(a,-,-)}(n) = q_{d}^{(a)}(n) - Q_{d}^{(a,-,-)}(n) \geq 0$ where $Q_{d}^{(a,-,-)}(n)$ counts the number of partitions into parts $\equiv \pm a \pmod{d + 3}$ excluding the $a$ and $d+3-a$ parts and proved it for infinitely many classes of $n$ and $d$. We prove that $Δ_{d}^{(3,-)}(n) \geq 0$ for all but finitely many $d$. We also provide a proof of the generalized conjecture for all but finitely many $d$ for fixed $a$ and strengthen the results of Duncan, Khunger, Swisher, and the second author. We provide a conditional proof of a linear lower bound on $d$ for the generalized conjecture, which improves our unconditional result based on a conjectural modification of a recently proven conjecture of Alder. Using this modification, we obtain a strengthening of this generalization of Kang and Park's conjecture which remarkably allows $a$ as a part. Additionally, we provide asymptotic evidence that this strengthened conjecture holds.
