We estimate the maximum of ${\prod }_{j=1}^n|1-x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_j$ is j.    In contrast we show, under fairly general conditions, that the maximum is less than $2^n/n^r$, where r is an arbitrary positive number. One consequence is that the...

A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial q-series pertaining to k-marked Durfee symbols, in which we allow additional singularities. We show that these...

Let $r(n,m)$ denote the number of partitions of $n$ into parts, each of which is at least $m$. By applying the saddle point method to the generating series, an asymptotic estimate is given for $r(n,m)$, which holds for $n\to \infty$, and $1\le m\le c_1{\displaystyle \frac{n}{{\left(logn\right)}^{c_2}}}$.

We present various results on the number of prime factors of the parts of a partition of an integer. We study the parity of this number, the extremal orders and we prove a Hardy-Ramanujan type theorem. These results show that for almost all partitions of an integer the sequence of the parts satisfies similar arithmetic properties as the sequence of natural numbers.

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t\ge 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$$4,$$6]. If $t\ge 1$ and $n\ge 0$, then we define...

Introduction. The problem of determining the formula for $P_S\left(n\right)$, the number of partitions of an integer into elements of a finite set S, that is, the number of solutions in non-negative integers, $h_{s₁},...,h_{s_k}$, of the equation hs₁ s₁ + ... + hsk sk = n, was solved in the nineteenth century (see Sylvester [4] and Glaisher [3] for detailed accounts). The solution is the coefficient of$xⁿin$[(1-xs₁)... (1-xsk)]-1, expressions for which they derived. Wright [5] indicated a simpler method by which to find part of the solution...

On désigne par $r(n,m)$ le nombre de partitions de l’entier $n$ en parts supérieures ou égales à $m$. En partant de l’estimation asymptotique de $r(n,m)$ exprimée à l’aide d’un paramètre $\sigma$ défini implicitement en fonction de $n$ et $m$, nous éliminons ce paramètre en utilisant la formule sommatoire d’Euler-Maclaurin, pour obtenir un développement asymptotique de $r(n,m)$ valable pour $n\to +\infty$, et $1\le m\le \Gamma \sqrt{n}$, $\Gamma$ étant un réel quelconque.

On désigne par $r(n,m)$ le nombre de partitions de l’entier $n$ en parts supérieures ou égales à $m$, et $R(n,m)=r(n-m,m)$ le nombre de partitions de $n$ de plus petite part $m$. Dans un précédent article (voir [9]) un développement asymptotique de $r(n,m)$ est obtenu uniformément pour $1\le m=O\left(\sqrt{n}\right)$ ; on complète ce développement uniformément pour $1\le m=\left(n{log}^{-3}n\right)$. Afin de prolonger les résultats jusqu’à $m\le n$, on donne un encadrement de $r(n,m)$ valable pour $n^{2/3}\le m\le n$ en utilisant la relation $r(n,m)={\sum }_{t=1}^{\lfloor n/m\rfloor }P(n-(m-1)t,t)$ où $P(i,t)$ désigne le nombre de partitions de $i$ en exactement $t$ parts. On donne aussi une...