An integer partition of $n$ is called graphical if its parts form a degree sequence of a simple graph. While unrestricted graphical partitions have been extensively studied, much less is known when the parts are restricted to a prescribed set. In this work, we investigate the probability that a uniformly random partition of an even integer $n$, subject to such restrictions, is graphical. We establish an upper bound on this probability expressed solely in terms of the Durfee square of the partition. Additionally, we evaluate the most-probable size of the Durfee square, depending solely on $n$ and the restrictions on the parts. Thus, we establish typical bounds on the probability that such random partitions are graphical. Our approach employs the Nash-Williams graphical condition, the saddle-point method and Edgeworth expansions.
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 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.
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.
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.
A new recursive procedure for calculation of restricted partition function is suggested. An explicit formula for the restricted partition function is found based on this procedure.
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 provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.
Using a new graphical representation for partitions, the author obtains a family of partition identities associated with partitions into distinct parts of an arithmetic progression, or, more generally, with partitions into distinct parts of a set that is a finite union of arithmetic progressions associated with a modular sum-free Sidon set. Partition identities are also constructed for sets associated with modular sum-free sets.
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…
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called Sylvester waves) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.
Given an integer base $b\geq 2$, a number $ρ\geq 1$ of colors, and a finite sequence $Λ=(λ_1,\ldots,λ_ρ)$ of positive integers, we introduce the concept of a $Λ$-restricted $ρ$-colored $b$-ary partition of an integer $n\geq 1$. We also define a sequence of polynomials in $λ_1+\cdots+λ_ρ$ variables, and prove that the $n$th polynomial characterizes all $Λ$-restricted $ρ$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.
In this paper, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our extended technique could be a useful tool to prove congruences for certain types of combinatorial functions based on a bounded number of calculations. As applications of our result, we establish new and existing restricted plane partition congruences, restricted plane overpartition congruences and several examples of restricted partition congruences.
In this paper, we investigate $π(m,n)$, the number of partitions of the \emph{bipartite number} $(m,n)$ into \emph{steadily decreasing} parts, introduced by L.Carlitz ['A problem in partitions', Duke Math Journal 30 (1963), 203--213]. We give a relation between $π(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, some uniform asymptotic formulas for $π(m,n)$ are established.
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…
The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves.
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 $\mathbb{Z}/n\mathbb{Z}$. .
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. We find the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method, where the saddle-point we need is associated to a zero of the analytically continued dilogarithm. Our main result disproves a conjecture of Rademacher.
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed.
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 …
Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank …
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this product as $N \to \infty$ by applying the saddle-point method to get the asymptotics of the main terms. In this second part we bound the error terms. This involves estimating products of sines and further saddle-point arguments. The saddle-points needed are associated to zeros of the analytically continued dilogarithm.
Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerator…
The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Raman…
In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent integers, and propose an analogue of Ramanujan's congruence relations for the `number of parts' function instead of the partition function. We also deduce the generating function for the `number of parts', and relate the technical results with their graphical interpretations through a …
For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for $p_{\mathcal A}(n)$, where $\mathcal A$ is the set of $k$-th powers of primes (for fixed $k$). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into $k$-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about …
The higher Sylvester waves are discussed. Techniques used involve finite difference operators. For example, using Herschel's theorem, elegant expressions for Euler's rational functions and the Todd operator are found. Derivative expansions are also rapidly treated by the same method. A general form for the wave is obtained using multiplicative series, and comments made on its further reduction. As is known, Dedekind sums arise in the case of coprime components and it is pointed out that Brioschi had this result, but not the terminology, very early on. His proof is repeated. Adding a set of one…
We study the correlators of irregular vertex operators in two-dimensional conformal field theory (CFT) in order to propose an exact analytic formula for calculating numbers of partitions, that is: 1) for given $N,k$, finding the total number $λ(N|k)$ of length $k$ partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$. 2) finding the total number $λ(N)=\sum_{k=1}^Nλ(N|k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $λ(N|k)$ by relating two-point short-distance correlation functions of irregular vertex operators in $c=1$ conformal field theory ( the …
