Related papers: General moment theorems for non-distinct unrestric…

Following the ideas of Rosenbloom [7] and Hayman [5], Luis B\'aez-Duarte gives in [1] a probabilistic proof of Hardy-Ramanujan's asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of…

For $k\geqslant 1$, denote by $p_k(n)$ the number of partitions of an integer $n$ into $k$-th powers. In this note, we apply the saddle-point method to provide a new proof for the well-known asymptotic expansion of $p_k(n)$. This approach…

For a positive integer $n$, let $p(n)$ be the number of ways to express $n$ as a sum of positive integers. In this note, we revisit the derivation of the Rademacher's convergent series for $p(n)$ in a pedagogical way, with all the details…

In this paper we strongly improve asymptotics for $s_1(n)$ (respectively $s_2(n)$) which sums reciprocals (respectively squares of reciprocals) of parts throughout all the partitions of $n$ into distinct parts. The methods required are much…

In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…

In this paper, we consider representations of integers as sums of at most four distinct $m$-gonal numbers (allowing a fixed number of repeats of each polygonal number occurring in the sum). We show that the number of such representations…

We propose an aproach for asymptotic analysis of plane partition statistics related to counts of parts whose sizes exceed a certain suitably chosen level. In our study, we use the concept of conjugate trace of a plane partition of the…

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n,…

A representation number is a function which expresses the number of ways an integer can be written as a sum of elements of chosen sets. One of the oldest number-theoretic results on representation numbers is Fermat's theorem which says that…

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of…

In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity.…

Using tools from representation theory, we derive expressions for the coincidence rate of partially-distinguishable particles in an interferometry experiment. Our expressions are valid for either bosons or fermions, and for any number of…

Let $p_{r,s}(n)$ denote the number of partitions of a positive integer $n$ into parts containing no multiples of $r$ or $s$, where $r>1$ and $s>1$ are square-free, relatively prime integers. We use classical methods to derive a…

We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum…

The minimal excludant statistic, which denotes the smallest positive integer that is not a part of an integer partition, has received great interest in recent years. In this paper, we move on to the smallest positive integer whose frequency…

We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then…

For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the…

We consider procedures of sampling parts from a random integer partition. We determine asymptotically the probabilty distribution of the randomly-selected part whenever the positive integer that is partitioned becomes large.