Fethi Ben Said, Jean-Louis Nicolas, Ahlem Zekraoui
Improving on some results of J.-L. Nicolas \cite {Ndeb}, the elements of the set ${\cal A}={\cal A}(1+z+z^3+z^4+z^5)$, for which the partition function $p({\cal A},n)$ (i.e. the number of partitions of $n$ with parts in ${\cal A}$) is even for all $n\geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this note we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews.
In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities.
Parity of parts and excludant statistics in partitions
本文研究了分拆中基于奇偶性的受限排除统计，利用q级数变换，证明其生成函数与量子模形式及相关函数有关，并通过Tauberian方法获得序列渐进行为。
We define an extension of parity from the integers to the rational numbers. Three parity classes are found -- even, odd and `none'. Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural density provides a means of distinguishing the sizes of countably infinite sets. The Calkin-Wilf tree has a remarkably simple parity pattern, with the sequence `odd/none/even' repeating indefinitely. This pattern means that the three parity classes have equal natural density in the rationals. A similar result holds for the Stern-Brocot tree.
We give a complete characterization of the parity of $b_8(n)$, the number of $8$-regular partitions of $n$. Namely, we prove that $b_8(n)$ is odd or even depending on whether or not we have the factorisation $24n+7=p^{4a+1}m^2$, for some prime $p\nmid m$ and $a\ge 0$.
Parity results of PEND partition
本文针对计数偶数部分不互异的n的分拆数的pend(n)集合，借助Newman的结果证明了其若干模2同余无穷族，为PEND分拆的奇偶性研究提供了新结论。
We improve S.-C. Chen's result on the parity of Schur's partition function. Let $A(n)$ be the number of Schur's partitions of $n$, i.e., the number of partitions of $n$ into distinct parts congruent to $1, 2 \mod{3}$. S.-C. Chen \cite{MR3959837} shows $\small \frac{x}{(\log{x})^{\frac{47}{48}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}$. In this paper, we improve Chen's result to $\frac{x}{(\log{x})^{\frac{11}{12}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}.$
We prove recent conjectures of Chern concerning nonnegativity of a certain q-series related to parity bias in integer partitions.
Combinatorial proof of an inequality on some partitions separated by parity
本文通过组合方法证明了关于按奇偶性分开的划分的不等式在n≥373时成立，回答了Bringmann等人的提问。
Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts
本文通过经典q级数操作和拉马努金θ函数性质，证明了关于特定算术级数中过着色划分函数的模2幂同余猜想。
We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of partitions of i into distinct parts with even rank minus the number with odd rank, for which there is an almost closed formula by Andrews, Dyson and Hickerson. This method of calculating the number of partitions of n with a given parity of the smallest part is practical and efficient.
Overcolored Partition $k$-tuples Restricted by Parity of the Parts
本文研究超着色分拆k元组的计数函数$\bar{b}^k_{r,s}(n)$，扩展Chacon与Sellers的相关结果，建立其模素数的可除性性质及模2幂的新同余式，所用技术基于theta函数恒等式和模形式。
On Partition Classes Arising from Parity, Differences, and Repeated Smallest Parts
本文研究了与奇偶性、差值和重复最小部分相关的划分类，推导了相关身份并扩展了欧拉划分定理，同时给出了组合和q-级数的证明。
Partitions with parity restrictions: a bijective approach
研究具有奇偶限制的整数划分，用双射方法证明相关恒等式，相比生成函数代数操作更简便。
Congruences for Overcolored Partition $k$-tuples Restricted by Parity of the Parts
本文提出统一框架研究过度着色分拆 $k$-元组 $\bar{b}^k_{r,s}(n)$ 模奇素数的同余性质，并证明特定情形下 $\bar{b}_{4,2}^4(5n)$ 模 5 同余为零。
Distributions of parity differences and biases in partitions into distinct parts
本文定义了分拆的奇偶差，证明了无重复部分分拆中归一化奇偶差数量的渐近展开式，并由此得出奇偶差与奇偶偏置的分布，同时推广至模 $N$ 的情形。
For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd pr…
Fixed Perimeter Analogues of Several Partition Results Related to Parity
该研究将奇偶性偏差不等式等分拆相关结果扩展至固定周长场景，推导了两类特殊分拆的递归公式，推广了经典分拆理论的相关结论。
A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of partitions with even crank $M_e(n)$ minus the number of partitions with odd crank $M_o(n)$ satisfies. For example, we show that $M_e(5n+4)-M_o(5n+4)\equiv 0 \pmod 5.$ We also determine the exact values of $M_e(n)-M_o(n)$ in case of partitions into distinct parts, which are at most two and zero for infinitely many $n$.
Parity Considerations for Mex-Related Partition Functions of Andrews and Newman
本文研究了Andrews和Newman提出的整数分拆mex相关函数族$p_{t,t}(n)$，并给出了$p_{1,1}(n)$和$p_{3,3}(n)$的完整奇偶性刻画。
Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions
研究Rogers-Ramanujan分拆中奇数秩与偶数秩超出数的生成函数，证明其与非Rascoe分拆（不同部分且部分数非部分）紧密相关，并推导算术性质及模4同余猜想，推广至参数ℓ。
On the asymptotic behavior for partitions separated by parity
本文运用 Ingham 的 Tauberian 定理，为 Andrews 在奇偶分隔分拆研究中提出的八个生成函数分别给出渐近主项。
Overcolored Partition Restricted by Parity of the Parts
本文扩展了Thejitha等人定义的a_{r,s}(n)函数，将其应用于过划分，考虑了部分奇偶性对多重着色划分的影响。
Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(λ)$ for an integer partition $λ$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $λ$. They then defined $p_{A,a}(n)$ to be the number of partitions $λ$ of $n$ satisfying $\text{mex}_{A,a}(λ)\equiv a\pmod{2A}$. They found the generating function for $p_{t,t}(n)$ and $p_{2t,t}(n)$ for any positive integer $t$, and studied their arithmetic properties for some small values of $t$. In this article, we study the partition function $p_{mt,t}(n)$ for all positive integers $m$ a…
Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the …
Proof of a conjecture of Andrews and El Bachraoui on the parity of two-color partitions
研究安德鲁斯和埃尔·巴赫拉维关于双色划分奇偶性的猜想，通过证明若\(q\)-级数傅里叶系数\(t_o(n)\)为奇数，\(8n + 9\)可由\(x^2 + 2y^2\)表示来进行验证。
The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions defined by Andrews. In particular, we prove that there are infinitely many congruences for $cϕ_k(n)$ modulo $\ell,$ where $\gcd(\ell,6k)=1,$ and we also prove results on the parity of $cϕ_k(n).$ Along the way, we prove results regarding the parity of coefficients of weakly holomorphic modular forms which generalize work of Ono.
Combinatorial proofs of inequalities involving the number of partitions with parts separated by parity
本文组合证明了具有按奇偶性分离部分的分拆数之间的几个不等式，并证明了 Fu 和 Tang 提出的一个关于部分重数受限分拆的猜想不等式。
Asymptotics of parity biases for partitions into distinct parts via Nahm sums
本文推广了Kim等人的奇偶偏差研究，利用圆法给出互异分拆的模N偏差渐近结果，同时提供了分析Nahm型q-超几何级数渐近性的独立有用技术。
A Goellnitz-Gordon partition is one in which the parts differ by at least 2, and where the inequality is strict if a part is even. Let Q_i(n) denote the number of partitions of n into distinct parts not congruent to i mod 4. By attaching weights which are powers of 2 and imposing certain parity conditions on Goellnitz-Gordon partitions, we show that these are equinumerous with Q_i(n) for i=0,2. These complement results of Goellnitz on Q_i(n) for i=1,3, and of Alladi who provided a uniform treatment of all four Q_i(n), i=0,1,2,3, in terms of weighted partitions into parts differing by >= 4. Our…
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p\geqslant 5$. In this paper, we introduce new infinite Ramanujan type congruences modulo $2$ for $b_3(2n)$. They complement naturally the results of Keith-Zanello and Yao and involve primes in $\mathcal P=\{p \text{ prime …
A combinatorial proof on partition function parity
该研究针对分拆函数奇偶性相关经典结论，通过构造双射与对合的组合方法，给出Subbarao强化结论的全新自包含证明，同时得到整数分拆特殊子集计数的更一般定理。
Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts
本文研究了受部分奇偶性限制的多色划分数目$a_{r,s}(n)$的算术性质，采用生成函数和$q$-级数技术进行分析。
We obtain some interesting results about the parity of the Fourier coefficients of hauptmoduln $j_{N}(z)$ and $j_{N}^{+}(z),$ for some positive integers $N$. We use elementary methods and the techniques of O. Kolberg's proof for the parity of the partition function.
The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely $p_{t,t}(n)$, and provide complete parity characterizations of $p_{1,1}(n)$ and $p_{3,3}(n)$. In this article, we study the parity of $p_{t,t}(n)$ when $t=2^α, 3\cdot 2^α$ for …
Modulo 2 congruences for partitions with initial repetitions
本文针对带初始重复的分拆类，推导了相关函数的奇偶性公式，并提出了无穷多组模2拉马努金型同余式，丰富了分拆同余理论。
Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of $a(2m)$ based solely on properties of $m.$ In this note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 …
If $ p_k(a,m,n) $ denotes the number of partitions of $n$ into $k$th powers with a number of parts that is congruent to $ a $ modulo $m,$ then $p_2(0,2,n)\sim p_2(1,2,n)$ and the sign of the difference $p_2(0,2,n)- p_k(1,2,n)$ alternates with the parity of $n,$ as proven by recent work of the author (2020). In this paper, we place the problem in a broader framework. By analytic arguments using the circle method and Gauss sums estimates, we show that the same results hold for any $ k\ge2. $ By combinatorial arguments, we show that the sign of the difference $p_k(0,2,n)- p_k(1,2,n)$ depends on t…
Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the combinatorial objects corresponding to a sum of positive terms, we establish bijections that lead a telescoping relation. We illustrate this idea by giving a combinatorial telescoping relation for a classical identity of MacMahon. Recently, Andrews posed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials which was derived bas…
Untrodden pathways in the study of partitions with parts separated by parity
本文研究Andrews引入的奇偶部分分隔分拆函数，证明其生成函数可表示为特定模形式，推导最小部分分拆函数的生成函数形式，给出新组合解释与量子q级数恒等式及渐近项。
Parity of the coefficients of certain eta-quotients, III: two special classes
本文研究了两类eta商的奇偶性，包括 Andrews 独特的奇异overpartitions 和纯eta幂的奇偶性，扩展了已知的密度定理。
Partitions with parts separated by parity: conjugation, congruences and the mock theta functions
本文利用共轭映射的组合方法研究奇偶部件分隔的整数分拆，证明推广 Andrews 二元生成函数的参数化恒等式与两族 Andrews–Beck 型同余式，并揭示共轭稳定分拆子集同三个三阶 mock theta 函数的联系。
We continue the study of the $(a,b,m)$-copartition function $\mathrm{cp}_{a,b,m}(n)$, which arose as a combinatorial generalization of Andrews' partitions with even parts below odd parts. The generating function of $\mathrm{cp}_{a,b,m}(n)$ has a nice representation as an infinite product. In this paper, we focus on the parity of $\mathrm{cp}_{a,b,m}(n)$. As with the ordinary partition function, it is difficult to show positive density of either even or odd values of $\mathrm{cp}_{a,b,m}(n)$ for arbitrary $a, b$, and $m$. However, we find specific cases of $a,b,m$ such that $\mathrm{cp}_{a,b,m}…
Explicit linear dependence congruence relations for the partition function modulo 4
针对分拆函数 p(n) 模 4 的线性同余依赖关系，本文首次显式构造了 Ono 理论证明存在的实例，揭示了其奇偶性并非完全随机。
In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}(n)$ and $p_{2t,t}(n)$ for all $t\geq 1$. Using these identities, we prove that the Ramanujan's famous congruences for $p(n)$ are also satisfied by $p_{t,t}(n)$ and $p_{2t,t}(n)$ for infinitely many values of $t$. Very recently, da Silva and Sellers …
A perimeter analogue of Franklin's identity and an inequality related to the parity of parts
本文通过生成函数与渐近分析，证明了关于分拆周长不等式的两个猜想，并建立了与无峰Motzkin路径的联系。
For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. For each $s\in\{1,3\}$, the generating function of $R^*_s(n)$, the number of partitions of $n$ into parts not congruent to $0$, $\pm s\pmod {10}$ and $10-2s \pmod {20}$, arises naturally in regime IV of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. In this paper, we investigate the parity of $R_s(n)$ and $R^*…
Sum of parts in overpartitions and partitions without repeated odd parts
本文通过经典 theta 函数恒等式，证明了按奇偶性分类的超划分中某些非划线部分和以及无重复奇数部分划分中某些部分和的模 5 和模 7 拉马努金型同余式。
Biases in Non-Unitary Partitions
本文研究了非单位ary划分中的奇偶偏差问题，证明了与已有结果类似的类比结论，并探讨了两种不同划分类别的不等式关系。
Parity of the partition function in quadratic progressions
本文证明了2010年提出的关于特定二次级数中分拆函数奇偶性包含无穷多奇偶项的猜想，通过几何方法结合伽罗瓦表示等完成证明，相关恒等式已在Lean中验证。
Convexity and log-concavity of the partition function weighted by the parity of the crank
本文利用 crank 奇偶分拆函数差的渐近公式及其显式界，证明 $M_0(n)$ 与 $M_1(n)$ 的凸性，并进一步得到其对数凹性及高阶 Turán 不等式，细化了经典分拆函数 $p(n)$ 的凸性结果。
Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
Overpartitions with parts separated by parity
本文研究了具有奇偶性间隔部分的超分拆的生成函数，并证明了与模形式、q-超几何级数和拟模形式相关的q-级数恒等式。
In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a relation between the rank of strongly unimodal sequences and the rank of integer partitions.
Rook decomposition of the Partition function
本文利用 Ferrers 棋盘的最大车数与 Durfee 三角形的联系，提出分拆函数的新车分解，并给出相关生成函数、模周期性、奇偶性及增长渐近结果。
A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$-series
本文证实了安德鲁斯和埃尔·巴赫拉维关于\(q\)级数\(T_o(q)\)系数奇偶性的猜想，还建立了\(S_1(q)\)系数模8的无穷多个同余式，并证明\(s_1(n)/8\)对\(n\geq0\)以自然密度1取整数值。
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
Define a "nuclear partition" to be an integer partition with no part equal to one. In this study we prove a simple formula to compute the partition function $p(n)$ by counting only the nuclear partitions of $n$, a vanishingly small subset by comparison with all partitions of $n$ as $n\to \infty$. Variations on the proof yield other formulas for $p(n)$, as well as Ramanujan-like congruences and an application to parity of the partition function.
An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for $t$-regular partitions, and determined conditions for when these functions are lacunary modulo powers of primes. We generalize their results in a number of ways by studying infinite products called Dedekind eta-quotients and generalized Dedekind eta-quotients. We then apply our results to the generating functions for the partition functions considered by Nekrasov, Ok…
We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all $m \leq 28$, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for ar…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。