Matthias Beck, Ira M. Gessel, Takao Komatsu
Given a set of positive integers A = {a_1,...,a_n}, we study the number p_A (t) of nonnegative integer solutions (m_1,...,m_n) to m_1 a_1 + ... m_n a_n = t. We derive an explicit formula for the polynomial part of p_A.
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Motivated by a question of Lovejoy \cite{lovejoy}, we show that three-colored Frobenius partition function $\c3$ and related arithmetic fuction $\cc3$ vanish modulo some powers of 5 in certain arithmetic progressions.
Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting combinatorial structure and led to proofs of a number of congruences. In this paper, we show how Andrews' generating functions can be cast in the framework of Eichler and Zagier's Jacobi forms. This reformulation allows us to compute explicit formulas for the generalized Frobenius partition generating functions (and in fact provides formulas for further functions…
Congruences modulo powers of $3$ for $6$-colored generalized Frobenius partitions
本文研究6色广义弗罗贝尼乌斯划分函数cϕ_6(n)的同余性质，采用Banerjee和Smoot发展的方法验证了Gu等人提出的猜想。
Partition-theoretic Frobenius-type limit formulas
本研究利用分拆生成函数技术，证明了推广复幂级数Abel收敛定理的Frobenius公式的q级数类似形式，明确了序列平均值存在时对应q级数极限与序列平均值的等价关系。
Let $k$ and $n$ be positive integers. Let $cϕ_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$ and $\mathrm{C}Φ_k(q)$ be the generating function of $cϕ_{k}(n)$. In this article, we study $\mathrm{C}Φ_k(q)$ using the theory of modular forms and discover new surprising properties of $\mathrm{C}Φ_k(q)$.
Let $a(n)$ defined by $\sum_{n=1}^{\infty}a(n)q^n := \prod_{n=1}^{\infty}\frac{1}{(1-q^{3n})(1-q^n)^3}.$ In this note, we prove that for every non-negative integer $n$, a(15n+6) \equiv 0\pmod{5}, a(15n+12) \equiv 0\pmod{5}. As a corollary, we obtained some results of Ono
We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite $q$-products. In particular, we show that specific examples of our result easily reestablish previously known formulas, and we describe new congruences, both conjectural and proven, in additional cases. The modular structure of Jacobi forms indicates t…
Congruences modulo powers of $3$ for generalized Frobenius partitions $CΨ_{6,0}$
本文研究了$(6,0)$彩色弗罗贝尼乌斯划分函数$cψ_{6,0}(n)$，通过阿特金-勒文纳逆运算建立生成函数关系，并证明其对3的幂次的同余式。
Let $cϕ_{k}(n)$ be the $k$-colored generalized Frobenius partition function. By employing the generating function of $cϕ_{6}(3n+1)$ found by Hirschhorn, we prove that $cϕ_{6}(27n+16)\equiv 0$ (mod 243). This confirms a conjecture of E.X.W. Xia. We also find a congruence relation $cϕ_{6}(81n+61) \equiv 3 cϕ_{6}(9n+7)$ (mod 243). Moreover, we show that $cϕ_{6}(81n+61) \equiv 0$ (mod 81), $cϕ_{6}(243n+142) \equiv 0$ (mod 243) and $cϕ_{6}(729n+ 547) \equiv 0$ (mod 243). We further conjecture that for $n\ge 0$, $cϕ_{6}(243n+142) \equiv 0$ (mod 729).
Let $N\geq 1$ be squarefree with $(N,6)=1$. Let $cϕ_N(n)$ denote the number of $N$-colored generalized Frobenius partition of $n$ introduced by Andrews in 1984. We prove $$ cϕ_N(n)= \sum_{d \mid N} N/d \cdot P\left( \frac{ N}{d^2}n - \frac{N^2-d^2}{24d^2} \right) + b(n)$$ where $C(z) := (q;q)^N_\infty\sum_{n=1}^{\infty} b(n) q^n$ is a cusp form in $S_{(N-1)/2} (Γ_0(N),χ_N)$. This extends and strengthens earlier results of Kolitsch and Chan-Wang-Yan treating the case when $N$ is a prime. As an immediate application, we obtain an asymptotic formula for $cϕ_N(n)$ in terms of the classical partiti…
Let $cϕ_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $cϕ_{3}(n)$ and $cϕ_{9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\ge 3$ and $n\ge 0$, we prove that \[cϕ_{3}\Big(3^{2k}n+\frac{7\cdot 3^{2k}+1}{8}\Big) \equiv 0 \pmod{3^{4k+5}}.\] We give two different proofs to the congruences satisfied by $cϕ_{9}(n)$. One of the proofs uses an relation between $cϕ_{9}(n)$ and $cϕ_{3}(n)$ due to Kolitsch, for which we provide a new proof in this paper.
We extend George Andrew's general principle for counting generalized Frobenius partitions to include arrays with nonzero row difference and establish some congruences for these arrays.
The Frobenius symbol was first introduced in 1900 by Frobenius as a way to encode an integer partition. In 1941, motivated by the modular representation theory of the symmetric group, Nakayama introduced the idea of a p-core partition, for p prime, using hook removals. In the following decade, Robinson, Littlewood, Staal and Farahat codified the $p$-quotient of such a partition, using variations of the star diagram. Since the 1970s, the convention has been to build up the theory of both core and quotients with the abacus construction, first introduced by G. James. In this paper we return to th…
The Littlewood decomposition via colored Frobenius partitions
本文利用Frobenius分拆给出Littlewood分解的另一种描述，并应用于自共轭分拆和双倍不同分拆，得到其$t$-核和$t$-商的新刻画。
Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions
本文研究k着色广义Frobenius分拆数的同余性质，证明了Cui等人关于cφ₁₈(3n+2)模2187为0的猜想，同时得到cφ₁₆(n)和cφ₁₈(n)的其他同余结果。
The generalized Frobenius problem via restricted partition functions
本文利用受限分拆函数的水平集与拟多项式渐近结构，研究广义 Frobenius 数及可表示整数计数，给出大 $k$ 时的公式、完整渐近性并恢复若干特例结果。
Correspondence among congruence families for generalized Frobenius partitions via modular permutations
本文通过构造向量值模形式，揭示了广义弗罗贝尼乌斯划分同余族之间的对应关系，并利用模变换证明了3色弗罗贝尼乌斯划分的同余关系。
In these two companion papers, we give infinite families of partition identities which generalise Primc's and Capparelli's identities, and study their consequences on the theory of crystal bases of the affine Lie algebra $A_{n-1}^{(1)}.$ In this first paper, we focus on combinatorial aspects. We give a $n^2$-coloured generalisation of Primc's identity by constructing a $n^2 \times n^2$ matrix of difference conditions, Primc's original identities corresponding to $n=2$ and $n=3$. While most coloured partition identities in the literature connect partitions with difference conditions to partitio…
Two-colored generalized Frobenius partitions and minimal-excludant sums over bipartitions
研究两种双色Frobenius划分的计数，通过组合证明将其与双划分的最小排除统计量之和及相等情况联系起来。
Andrews, Brietzke, Rødseth and Sellers proved an infinite family of congruences on the number of the restricted $m$-ary partitions when $m$ is a prime. In this note, we show that these congruences hold for arbitrary positive integer $m$ and thus confirm the conjecture of Andrews, et al.
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".
We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d. We introduce a method to compute upper bounds for g(a_1,a_2,a_3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.
On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$
本文证明了对所有正整数 $\ell$、素数 $p\geq 5$ 及满足 $24r+1$ 为模 $p$ 二次非剩余的 $r$，广义 Frobenius 分拆函数 $\phi_{p\ell-1}(pn+r)$ 恒为偶数，给出无限族 $k$ 的奇偶性结果，证明基于初等生成函数操作。
Congruences modulo powers of $2$ for three restricted partition functions of Pushpa and Vasuki
本文通过q级数技术，研究了三项受限分割函数对2的幂次的同余关系，并推广了之前发现的特定同余关系。
Connections between colored restricted $b$-ary and ordinary partitions
该研究建立了带色有界普通分拆与带色二元、$b$元分拆间的联系，基于特殊递推关系得到带色$b$元分拆的恒等式及合成分拆相关结果。
Inequalities and asymptotics for hook numbers in restricted partitions
本文研究了受限划分中钩数的渐进行为，通过渐近公式证明了奇数部分划分在平均意义上比不同部分划分有更多钩数。
We derive the asymptotic formula for $p_n(N,M)$, the number of partitions of integer $n$ with part size at most $N$ and length at most $M$. We consider both $N$ and $M$ are comparable to $\sqrt{n}$. This is an extension of the classical Hardy-Ramanujan formula and Szekeres' formula. The proof relies on the saddle point method.
Congruence families modulo powers of $7$ for $4$-colored generalized Frobenius partitions
本文运用Chen-Zhu模变换框架研究4色广义Frobenius分拆$cψ_{4,β}$，给出模7的幂的Ramanujan型同余族新例证，并显著简化了证明。
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。
Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function $$p_{A(n)}(m(n)):=#\{(x_1,...,x_s)\in \mathbb{Z}^{s}: all x_j\geqslant 0, x_1a_1(n)+...+x_sa_s(n)=m(n) \}$$ when $s=2$ or 3. In either case, the formula implies that the function is an integer-valued quasi-polynomial. The main result is proved by a reciprocity law for a class of fractional part sums and the theory of generalized Euclidean division.
Identities involving partitions with distinct even parts and $4$-regular partitions
本文在已知不同偶数部分分拆数与4-正则分拆数相等的基础上，证明了一类受限的不同偶数部分分拆与受限4-正则分拆之间的恒等式。
Generalized Frobenius Partitions Modulo Powers of $2$
本文证明广义Frobenius分拆数模$2$的幂的同余式，完全确定$c\phi_{18}(2n+1)$模$16$和$c\phi_{10}(2n+1)$模$8$的值，并证明Das等人猜想。
We narrow in on the number of graphical partitions for which there is no known generating function by manipulating the well known generating function for Frobenius partitions.
Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. Let $p_\mathcal{A}(n,k)$ denote the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$. We investigate periodicity of the sequence and lower and upper bounds for the density of the set $\{n\in\mathbb{N}: p_\mathcal{A}(n,k) \equiv i \pmod{m}\}$ for a fixed positive integer $k$ and $i\in\{0,1,\ldots, m-1\}$. In particular, we apply our results t…
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 …
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 derive an explicit formula for a restricted partition function P_n^m(s) with constraints making use of known expression for a restricted partition function W_m(s) without constraints
Restricted partition functions and the $r$-log-concavity of quasi-polynomial-like functions
该研究针对弱递增正整数序列对应的受限分拆函数，推广其对数凹性性质，解决了强对数凹性及类拟多项式函数与受限分拆函数的r-对数凹性问题，给出有效解法。
On Graphical Partitions with Restricted Parts
研究受限图划分的概率，通过Durfee平方建立上界，并证明概率趋于零，结合Nash-Williams条件、乘法方法和Edgeworth展开。
A note on congruences modulo 7 and 11 for two restricted partition functions
本文通过q级数技术研究了某些受限划分函数在7和11模下的同余关系，并推广了之前使用模形式方法得到的结果。
In this note, we provide a simple derivation of expressions for the restricted partition function and its polynomial part. Our proof relies on elementary algebra on rational functions and a lemma that expresses the polynomial part as an average of the partition function.
We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for o…
Algebraic Structures on Sets of Partitions
受Robert Schneider工作的启发，本文探索分拆上的二元运算，发现受限分拆集合构成阿贝尔群、向量空间和交换环，为分拆分析提供代数工具。
We use the 0-1 tableaux as a tool for enumerating permutations and partitions with restricted minima. The method used is extended for permutation pairs and partition pairs generated by a bipartite 0-1 tableaux.
Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset $\{a_1,a_2,\ldots,a_k\}$. We find out a new type of Bessenrodt-Ono inequality for the function $p_\mathcal{A}(n,k)$. Further, we discover when and under what conditions on $k$, $\{a_1,a_2,\ldots,a_k\}$ and $N\in\mathbb{N}_+$, the sequence $\left(p_\mathcal{A}(n,k)\right)_{n=N}^\infty$ is log-concave. Our proofs are based on the asymptotic behavior of $p_\mathcal{A}(n,k)$, i…
Let $\mathbf a=(a_1,\ldots,a_r)$ be a vector of positive integers. In continuation of a previous paper we present other formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$.
Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of the mex) by proving a number of properties that naturally relate these partition statistics.
Remarks on a certain restricted partition function of Lin
本文通过q级数和分割公式，给出了林关于b(3n+1)和b(3n+2)生成函数的初等证明，并建立了b(n)对3的无限多个内部同余式。
Based on discrete truncated powers, the beautiful Popoviciu's formulation for restricted integer partition function is generalized. An explicit formulation for two dimensional multivariate truncated power functions is presented. Therefore, a simplified explicit formulation for two dimensional vector partition functions is given. Moreover, the generalized Frobenius problem is also discussed.
In the paper, the authors present several new relations and applications for the combinatorial sequence that counts the possible partitions of a finite set with the restriction that the size of each block is contained in a given set. One of the main applications is in the study of lonesum matrices.
For a vector $\mathbf a=(a_1,\ldots,a_r)$ of positive integers we prove formulas for the restricted partition function $p_{\mathbf a}(n): = $ the number of integer solutions $(x_1,\dots,x_r)$ to $\sum_{j=1}^r a_jx_j=n$ with $x_1\geq 0, \ldots, x_r\geq 0$ and its polynomial part.
Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions.
We study filters in the partition lattice formed by restricting to partitions by type. The Möbius function is determined in terms of the easier-to-compute descent set statistics on permutations and the Möbius function of filters in the lattice of integer compositions. When the underlying integer partition is a knapsack partition, the Möbius function on integer compositions is determined by a topological argument. In this proof the permutahedron makes a cameo appearance.
We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one irreducible and one homogeneous numerical semigroup at each class in the relation, and from those two elements we reconstruct the whole class. An alternative more efficient method is proposed based on the use of the Kunz-coordinates vectors of the elements in S(F).
The Li--Wan sieve is extended to multisets when the underlying set is symmetric. The main ingredient of the proof is the Mobius inversion formula on the poset of partitions of $\{1,2,\dots,k\}$ ordered by refinement. As illustrative applications, we investigate the problems of partitions over finite fields and zero-sum multisets over the additive group $\mathbb{Z}/n\mathbb{Z}$. .
A Combinatorial Approach to Frobenius Numbers of Some Special Sequences (Complete Version)
本文提出一种新的组合方法来解决弗罗贝尼乌斯问题，通过将问题转化为更易处理的优化问题，推导出弗罗贝尼乌斯数、赛尔伯格数和赛尔伯格和的公式。
We prove a conjecture due independently to Yan and Martinez--Savage that asserts inversion sequences with no weakly decreasing subsequence of length $3$ and enhanced $3$-noncrossing partitions have the same cardinality. Our approach applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-Mélou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced $3$-noncrossing partitions.
We generalize overpartitions to (k,j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems: generating function dissections, modular forms, bijections, and other combinatorial maps. We find connections to divisor sums, the Han/Nekrasov-Okounkov hook length formula and a possible approach to a finitization, and other topics, suggesting that a rich mine of results is available.
For a set of positive integers $A$, let $p_A(n)$ denote the number of ways to write $n$ as a sum of integers from $A$, and let $p(n)$ denote the usual partition function. In the early 40s, Erdős extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $α$ if and only if $\log p_A(n) \sim \log p(αn)$. Nathanson asked if Erdős's theorem holds also with respect to $A$'s lower density, namely, whether $A$ has lower-density $α$ if and only if $\log p_A(n) / \log p(αn)$ has lower limit $1$. We answer this question negatively by constructing, for every $α> 0$, a set …
We consider $(k,j)$-colored partitions, partitions in which $k$ colors exist but at most $j$ colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where $k$ and $j$ are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the $N \times M$ box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.
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