Related papers: Partitions into Piatetski-Shapiro sequences

We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise…

Assume that $m,s\in\mathbb N$, $m>1$, while $f$ is a polynomial with integer coefficients, $\text{deg}~f>1$, $f^{(i)}$ is the $i$th iteration of the polynomial $f$, $\kappa_n$ has a discrete uniform distribution on the set $\{0,1,\ldots,m^n…

We prove asymptotic formulas for the complex coefficients of $(\zeta q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to determine secondary terms in the asymptotics for $p(a,b,n)$, the number of integer partitions…

In this paper we study the Product Partition Problem (PPP), i.e. we are given a set of $n$ natural numbers represented on $m$ bits each and we are asked if a subset exists such that the product of the numbers in the subset equals the…

Let $\alpha_k(\lambda)$ denote the number of $k$-hooks in a partition $\lambda$ and let $b(n,k)$ be the maximum value of $\alpha_k(\lambda)$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $b(n,1)$. We…

For positive integers $m$ and $n$, the partial permutohedron $\mathcal{P}(m,n)$ is a certain integral polytope in $\mathbb{R}^m$, which can be defined as the convex hull of the vectors from $\{0,1,\ldots,n\}^m$ whose nonzero entries are…

In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of…

The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. It is expected that there are infinitely many primes in a Piatetski-Shapiro…

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at…

Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\geq 3$. One forms a subset of the set of $p$-strict partitions, and the other forms that…

In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the…

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of…

Motivated by the theorem of Gy\H ori and Lov\'asz, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$…

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be…

Let ${p}_{3,3}(n)$ denote the number of $2$-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $3$. In this paper, we shall establish some interesting Ramanujan-type congruences for…

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…

Let $\overline{p}_o(n)$ denote the number of overpartitions of $n$ into odd parts. The partition function $\overline{p}_o(n)$ has been the subject of many recent studies where many explicit Ramanujan-like congruences were discovered. In…

In this article we extend results of Kakutani, Adler-Flatto, Smilansky and others on the classical $\alpha$-Kakutani equidistribution result for sequences arising from finite partitions of the interval. In particular, we describe a…

A partition is a $t$-core partition if $t$ is not one of its hook lengths. Let $c_t(N)$ be the number of $t$-core partitions of $N$. In 1999, Stanton conjectured $c_t(N) \le c_{t+1}(N)$ if $4 \le t \ne N-1$. This was proved for $t$ fixed…

Let $f:\mathbb{N}\to\{0,\pm 1\}$, for $n \in \mathbb{N}$ let $\Pi[n]$ be the set of partitions of $n$, and for all partitions $\pi = (a_1,a_2,\ldots,a_k) \in \Pi[n]$ let \[ f(\pi) := f(a_1)f(a_2) \cdots f(a_k). \] With this we define the…