17 pages, 2 figures, to appear in Journal of Number Theory Updated to accepted version. Removed previously known results about self-conjugate 7-cores and corrected other historical information
Stanton [Open positivity attributes for integer partitions，Trends Math.，2：19-25，1999]已经成为最近对T-核分区的许多研究的催化剂。我们猜想斯坦顿一样的单调性结果比较自共轭（t+2）和t-核心分区n。我们得到的部分结果对这些aesturtures的值的t是大的，相对于n，和应用程序块理论的对称和交替的群体。为此，我们证明了公式的自共轭的T-核心分区的数量n作为一个函数的自共轭分区的数量较小的n。此外，我们还讨论了自共轭6-核心分区的积极性，并介绍了未来的研究领域，在表示论，渐近分析，单峰性，和数值恒等式和不等式。
英文摘要：
A conjecture on the monotonicity of t-core partitions in an article of Stanton [Open positivity conjectures for integer partitions, Trends Math., 2:19-25, 1999] has been the catalyst for much recent research on t-core partitions. We conjecture Stanton-like monotonicity results comparing self-conjugate (t+2)- and t-core partitions of n.
We obtain partial results toward these conjectures for values of t that are large with respect to n, and an application to the block theory of the symmetric and alternating groups. To this end we prove formulas for the number of self-conjugate t-core partitions of n as a function of the number of self-conjugate partitions of smaller n. Additionally, we discuss the positivity of self-conjugate 6-core partitions and introduce areas for future research in representation theory, asymptotic analysis, unimodality, and numerical identities and inequalities.
Partition theory abounds with bijections between different types of partitions. One of the most famous partition bijections maps each self-conjugate partition of a positive integer $n$ to a partition of $n$ into distinct odd parts, and vice versa. Here we prove new necessary and sufficient conditions for a self-conjugate partition to be $t$-core, in terms of only the parts of the corresponding partition into distinct odd parts, by proving a new hook length formula. Corollaries of these results include new applications of $t$-core self-conjugate partitions to subsets of the natural numbers, due…
We extend recent results of Ono and Raji, relating the number of self-conjugate $7$-core partitions to Hurwitz class numbers. Furthermore, we give a combinatorial explanation for the curious equality $2\operatorname{sc}_7(8n+1) = \operatorname{c}_4(7n+2)$. We also conjecture that an equality of this shape holds if and only if $t=4$, proving the cases $t\in\{2,3,5\}$ and giving partial results for $t>5$.
Institut Camille Jordan, Université Claude Bernard Lyon 1（里昂第一大学卡米耶·若当研究所）；Univ. de Tours, UMR CNRS 7013, Institut Denis Poisson（图尔大学，CNRS联合研究实验室7013，丹尼斯·普瓦松研究所）
We classify the connection between $t$-cores and self-conjugate $t$-cores to sums of squares. To do so, we provide explicit maps between $t$-core partitions and self-conjugate $t$-core partitions of a positive integer $n$ to representations of certain numbers as sums of squares. For example, the self-conjugate $4$-core partition $λ=(4,1,1,1)$ corresponds uniquely to the solution $61=6^2+5^2$. As a corollary, we completely classify the relationship between $t$-cores and Hurwitz class numbers. Using these tools, we see how certain sets of representations as sums of squares naturally decompose in…
We exactly determine the maximum possible hook length of $(s,t)$-core partitions with $d$-distinct parts when there are finitely many such partitions. Moreover, we provide an algorithm to construct a $d$-distinct $(s,t)$-core partition with this maximum possible hook length.
Refined counting of core partitions into $d$-distinct parts
使用组合双射和某些算盘图，Nath和Sellers已经将$(s, m s \pm 1)$ -核心分区枚举为不同的部分。我们推广他们的结果在几个方向，包括这些分区的部分的数量，通过考虑$d$ -不同的分区，并允许更一般的$(s, m s \pm r)$ -核心分区。作为我们的方法的一个应用程序，我们得到的平均和最大数量的部分，这些核心分区。
This paper investigates partitions which have neither parts nor hook lengths divisible by $p$, referred to as $p$-core $p'$-partitions. We show that the largest $p$-core $p'$-partition corresponds to the longest walk on a graph with vertices $\{0, 1, \ldots, p-1\}$ and labelled edges defined via addition modulo $p$. We also exhibit an explicit family of large $p$-core $p'$-partitions, giving a lower bound on the size of the largest such partition which is of the same degree as the upper bound found by McSpirit and Ono.
For primes $p$, we study the maximal possible size of a $p$-core $p'$-partition (a partition with no hook lengths or parts divisible by $p$). McDowell recently proved that the maximum is attained by a unique partition, say $Λ_p$. Using his graph theoretic description of $Λ_p$, we prove for $p > 10^6$ that \[\frac{1}{24}p^6 - p^5\sqrt{p} < |Λ_p| < \frac{1}{24}p^6 - \frac{1}{200}p^5\sqrt{p},\] which shows that $|Λ_p| \sim p^6/24$ as $p \to \infty$.
This note is concerned with the set of integral solutions of the equation $x^2+3y^2=12n+4$, where $n$ is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the 3-core partitions of n. As a consequence we describe an explicit bijection betwe…
Arithmetic Identities and Congruences for Partition Triples with 3-cores
令${{B}_{3}}(n)$表示$n$的分区三元组的数量，其中每个分区是3核的。借助于生成函数的操作，我们找到了${{B}_{3}}(n)$的算术恒等式和同余式的几个无穷族。另外，设$ω(n)$表示非负整数$n$的表示数，形式为$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$，我们发现了${{B}_{3}}(n)$与$ω(n)$之间的三种算术关系，如$ω(6n+5)=4{{B}_{3}}(6n+4).$
In this work, we analyze the behavior of the self-conjugate 6-core partition numbers $sc_{6}(n)$ by utilizing the theory of quadratic and modular forms. In particular, we explore when $sc_{6}(n) > 0$. Positivity of $sc_{t}(n)$ has been studied in the past, with some affirmative results when $t > 7$. The case $t = 6$ was analyzed by Hanusa and Nath, who conjectured that $sc_{6}(n) > 0$ except when $n \in \{2, 12, 13, 73\}$. This inspires a theorem of Alpoge, which uses deep results from Duke and Schulze-Pillot to show that $sc_{6}(n) > 0$ for $n \gg 1$ using representation numbers of a particul…
We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where $p$ is a prime.
Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally…
The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an ef…
Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$.
A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result, both the authors \cite{MeherJindal2022} recently proved that for a non-negative integer $α,$ $a_{3^α m}(n)$ is almost always divisible by arbitrary power of $2$ and $3$ and $a_{t}(n)$ is almost always divisible by arbitrary power of $p_i^j,$ where $j$ is a fixed positive integer and $t= p_1^{a_1}p_2^{a_2}\ldots p_m^{a_m}$ with primes $p_i \geq 5.$ In this art…
A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Recently, both authors \cite{MeherJindal2022} proved density results for $a_3(n)$, wherein we proved that $a_3(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ In this article, we prove that for a non-negative integer $α,$ $a_{3^α m}(n)$ is almost always divisible by arbitrary power of $2$ and $3.$ Further, we prove that $a_{t}(n)$ is almost always divisible by arbitr…
