Related papers: On partitions of $\mathbb{Z}_{m}$ with the same re…
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…
We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c}…
We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum\limits_{n=0}^{p-1}\dbinom{2n}{n} \equiv\eta_{p}\mod p^{2},$$ $$\sum\limits_{n=0}^{rp-1}\dbinom{2n}{n}…
Motivated by Andrews' partitions with initial repetitions, we derive parity formulas for several functions for this class of partitions. In many cases, we present an infinite family of Ramanujan-like congruences modulo 2.
In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible…
The partition function $p(n)$ and many of its related restricted partition functions have recently been shown independently to satisfy log-concavity: $p(n)^2 \geq p(n-1)p(n+1)$ for $n\geq 26$, and satisfy the inequality: $p(n)p(m) \geq…
Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural…
Let $M$ be a positive integer and $p(n)$ be the number of partitions of a positive integer $n$. Newman's Conjecture asserts that for each integer $r$, there are infinitely many positive integers $n$ such that \[ p(n)\equiv r \pmod{M}. \]…
We find a closed-form solution to the recurrence $\ z_{n+2} = \frac{1}{z_{n+1}} + z_n$, where $n \in \mathbb Z_{\geq 1}$ and $z_1 \in \mathbb R_{> 0},\ z_2 \in \mathbb R_{> 0}$. As a corollary, we derive an alternate proof of a recurrence…
We give three proofs of the following result conjectured by Carriegos, De Castro-Garc\'{\i}a and Mu\~noz Casta\~neda in their work on enumeration of control systems: when $\binom{k+1}{2} \le n < \binom{k+2}{2}$, there are as many partitions…
In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$…
We study the way in which the abstract structure of a small overlap monoid is reflected in, and may be algorithmically deduced from, a small overlap presentation. We show that every C(2) monoid admits an essentially canonical C(2)…
We consider representations of general non-overlapping placements of rectangles by spatial relations (west, south, east, north) of pairs of rectangles. We call a set of representations complete if it contains a representation of every…
For an integer $m\ge 2$, a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $\lambda_i$'s form a nonincreasing sequence. We extend a bijection…
In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences' terms, our problem can be really difficult, which calls for novel…
We prove that $K_{n}$ is $\mathbb{Z}_{2}^{2}$-cordial if and only if $1 \leq n \leq 3$ and that $K_{m,n}$ is $\mathbb{Z}_{2}^{2}$ if and only if it is false that $m=n=2$.
We consider Nahm's equations on a bounded open interval with first order poles at the ends. By imposing further boundary conditions, extended moduli spaces are identified with spaces of rational maps from $\mathbb{C}\mathrm{P}^1$ to…
We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^{\alpha}\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number…
The goal of this note is to provide an alternative proof of Theorem 1.1 (i) in [4], that is, if $n\geq 2$ and $M^{\alpha}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for some $\alpha\in \mathbb{C}$ and $p\geq 2$, then we have \begin{align*}…