Related papers: On the asymptotics of higher-dimensional partition…

In this paper, we enumerate Newton polygons asymptotically. The number of Newton polygons is computable by a simple recurrence equation, but unexpectedly the asymptotic formula of its logarithm contains growing oscillatory terms. As the…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

Let $F_n(x)$ be the partition polynomial $\sum_{k=1}^n p_k(n) x^k$ where $p_k(n)$ is the number of partitions of $n$ with $k$ parts. We emphasize the computational experiments using degrees up to $70,000$ to discover the asymptotics of…

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work…

We obtain an asymptotic expansion for $p(n)$, the number of partitions of a natural number $n$, starting from a formula that relates its generating function $f(t), t\in (0,1)$ with the characteristic functions of a family of sums of…

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We…

Let m be a positive integer, and let A be the set of all positive integers that belong to a union of r distinct congruence classes modulo m. We assume that the elements of A are relatively prime, that is, gcd(A) = 1. Let p_A(n) denote the…

We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition…

Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).

We will prove an infinite family of asymptotic formulas for the logarithm of certain two-colored partitions. An infinite sub-family of these asymptotics was posed as a conjecture by Guadalupe.

In this paper, we consider the asymptotic properties of hook numbers of partitions in restricted classes. More specifically, we compare the frequency with which partitions into odd parts and partitions into distinct parts have hook numbers…

Here we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion…

We deduce from the strong form of the Hardy--Ramanujan asymptotics for the partition function $p(n)$ an asymptotics for $p_{-S}(n)$, the number of partitions of $n$ that do not use parts from a finite set $S$ of positive integers. We apply…

We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions.…

The article motivates, presents and describes large computer calculations concerning the asymptotic behaviour of arithmetic properties of coefficient fields of modular forms. The observations suggest certain patterns, which deserve further…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in…

We construct an asymptotic approximation to the solution of a transmission problem for a body containing a region occupied by many small inclusions. The cluster of inclusions is characterised by two small parameters that determine the…

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed $3$-manifold groups…