Given relatively prime positive integers, $a_1,\ldots,a_n$, the Frobenius number is the largest integer with no representations of the form $a_1x_1+\cdots+a_nx_n$ with nonnegative integers $x_i$. This classical value has recently been generalized: given a nonnegative integer $k$, what is the largest integer with at most $k$ such representations? Other classical values can be generalized too: for example, how many nonnegative integers are representable in at most $k$ ways? For sufficiently large $k$, we give formulas for these values by understanding the level sets of the restricted partition function (the function $f(t)$ giving the number of representations of $t$). Furthermore, we give the full asymptotics of all of these values, as well as reprove formulas for some special cases (such as the $n=2$ case and a certain extremal family from the literature). Finally, we obtain the first two leading terms of the restricted partition function as a so-called quasi-polynomial.
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.
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.
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…
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…
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…
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…
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.
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.
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.
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 …
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
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…
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.
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 $\mathbb{Z}/n\mathbb{Z}$. .
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.
