Related papers: Partition regularity of generalized Pythagorean pa…
We generalize overpartitions to (k,j)-colored partitions: k-colored partitions in which each part size may have at most j colors. We find numerous congruences and other symmetries. We use a wide array of tools to prove our theorems:…
In a recent paper, Thejitha and Fathima introduced the overcolored partition function $\bar{a}_{r,s}(n)$, which enumerates overpartitions in which even parts may appear in one of $r$ colors and odd parts in one of $s$ colors, for fixed…
A quadratic polynomial $\Phi_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $\Phi_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if…
In 2000, A. Tripathi used generating functions to obtain a formula for the number of non-negative solutions (x,y) of the equation ax + by = n where a, b and n are given positive integers. We generalize this procedure for the number of…
We consider the simultaneous Pell equations $$x^2 - ay^2 = 1, \qquad z^2 - bx^2 = 1,$$ where $a > b\geq 2$ are positive integers. We describe a procedure which, for any fixed $b$, either confirms that the simultaneous Pell equations have at…
We consider the number of the $6$-regular partitions of $n$, $b_6(n)$, and give infinite families of congruences modulo $3$ (in arithmetic progression) for $b_6(n)$. We also consider the number of the partitions of $n$ into distinct parts…
We give a systematic study of q-algebraic equations. The questions of existence, uniqueness and regularity of the solutions are solved in the space of grid-based Hahn series. The regularity is understood in terms of asymptotic behavior of…
In this note we conjecture Rogers-Ramanujan type colored partition identities for an array with odd number of rows w such that the first and the last row consist of even positive integers. In a strange way this is different from the…
A system of linear equations in $\mathbb{F}_p^n$ is \textit{common} if every two-colouring of $\mathbb{F}_p^n$ yields at least as many monochromatic solutions as a random two-colouring, asymptotically as $n \to \infty$. By analogy to the…
Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…
Edge-matching problems, also called edge matching puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem…
Partitions of the set of primes are introduced based on the Chebyshev polynomials at rationals. The prime densities of all such partitions are established. Euler's Criterion for $SL(2,\mathbb Q)$ is formulated, which is the bridge between…
$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal…
We consider $(k,j)$-colored partitions, partitions in which $k$ colors exist but at most $j$ colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including…
About thirty years ago we looked for "minimal assumptions" on the data which guarantee that solutions to the $\,2-D\,$ evolution Euler equations in a bounded domain are classical. Classical means here that all the derivatives appearing in…
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients…
We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A…
A matrix A is image partition regular over Q provided that whenever Q - {0} is finitely coloured, there is a vector x with entries in Q - {0} such that the entries of Ax are monochromatic. It is kernel partition regular over Q provided that…
In this paper, we obtain pointwise convergence of solutions to the Schrodinger equation along a class of curves in $\mathbb{R}^{2}$ by the polynomial partitioning.