相关论文: A proof of Andrews' conjecture on Partitions with …

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties,…

We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect…

A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the…

We consider the problem of sequentially testing a simple null hypothesis versus a composite alternative hypothesis that consists of a finite set of densities. We study sequential tests that are based on thresholding of mixture-based…

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the…

In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…

Consider a system \Psi of non-constant affine-linear forms \psi_1,...,\psi_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of…

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…

Holroyd and Propp used Hall's marriage theorem to show that, given a probability distribution pi on a finite set S, there exists an infinite sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the number of i in…

Partitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition…

There are $n$ independent Bernoulli random variables $I_{k}$ with parameters $p_{k}$ that are observed sequentially. We consider a generalization of the Last-Success-Problem considering $w_{k}$ positive payments if the player successfully…

We establish the asymptotic behavior of the Andrews $G_k(q)$ function as $q\to 1.$

For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r…

In 1971 Cusick proved that every real number $x\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \{ x\in[0,1] : a_n(x) \geq k \text{ for all }…

Folsom, Males, Rolen, and Storzer recently proved Andrews' Conjecture~4 for the coefficients of \[ v_1(q)=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}=\sum_{n\ge 0}V_1(n)q^n. \] They also proved a refined density-one version of Andrews'…

We show that the universally axiomatized, induction-free theory PA^- is a sequential theory in the sense of Pudl\'ak [5], in contrast to the closely related Robinson's arithmetic.

Given two sets of positive integers $A$ and $B$, let $AB := \{ab : a \in A,\, b \in B\}$ be their product set and put $A^k := A \cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every…

Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $\lambda$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(\lambda)$ be a positive rth difference chosen uniformly at random in…

Aldous' spectral gap conjecture, proven by Caputo, Liggett and Richthammer, states the following: for any set of transpositions in the symmetric group $\mathrm{Sym}(n)$, the spectral gap of the corresponding random walk on the group -- an…

We prove two basic conjectures on the distribution of the smallest singular value of random n times n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n^{-1/2}, which…