Related papers: Overpartitions and Bressoud's conjecture, II

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent…

In this paper, we present a new Rogers--Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur's theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial…

Assuming that Brouwers Conjecture the upper bound for the sum of t< n largest eigenvalues of Laplacian graph on n vertices true for n <n_0, we prove the Brouwers Conjecture BC for n > n_0 for some fixed n_0

We solve multiple conjectures by Byszewski and Ulas about the sum of base $b$ digits function. In order to do this, we develop general results about summations over the sum of digits function. As a corollary, we describe an unexpected new…

In this paper, we generalize recent work of Mizuhara, Sellers, and Swisher that gives a method for establishing restricted plane partition congruences based on a bounded number of calculations. Using periodicity for partition functions, our…

Zeros of Bessel functions $J_\alpha$ play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over $\overline{\mathbb{Q}}$, and more generally of $E$-functions. In this paper we…

In this paper, we prove a conjecture proposed by George Beck, which involves gap-free partitions and partitions with distinct parts.

We show that the number of partitions of n with alternating sum k such that the multiplicity of each part is bounded by 2m+1 equals the number of partitions of n with k odd parts such that the multiplicity of each even part is bounded by m.…

Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain…

In this paper, we are motivated by the conjectures proposed by C.~Bender \textit{et al.}, \cite{C} in 2024. We have settled the first two conjectures negatively by providing a counter example in \cite{KTJ}, whereas in this paper, we prove…

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu'…

The BK inequality (\cite{BK85}) says that,for product measures on $\{0,1\}^n$, the probability that two increasing events $A$ and $B$ `occur disjointly' is at most the product of the two individual probabilities. The conjecture in…

We use the $q$-binomial theorem, the $q$-Gauss sum, and the ${}_2\phi_1 \rightarrow {}_2\phi_2$ transformation of Jackson to discover and prove many new weighted partition identities. These identities involve unrestricted partitions,…

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…

We present an involution on set partitions that interchanges two statistics related to relative size of block entries and use it to establish an equidistribution on objects counted by the Bessel numbers.

In this paper we give a complete proof of the Brumer-Stark conjecture over $\mathbf{Z}$.

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof…

Let $N(\leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(\leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(\leq m,n)\leq…