Related papers: Arithmetic properties for generalized cubic partit…

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton…

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…

In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if $M\geq 5$ is an integer and the integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$, and the…

In 2019, Andrews investigated integer partitions in which all parts of a given parity are smaller than those of the opposite parity and introduced eight partition functions based on the parity of the smaller parts and parts of a given…

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…

A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of…

We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.

This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…

The Hardy-Ramanujan formula for the number of integer partitions of $n$ is one of the most popular results in partition theory. While the unabridged final formula has been celebrated as reflecting the genius of its authors, it has become…

We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

It is proved that if we partition a $d$-dimensional cube into $n^d$ small cubes and color the small cubes into $m+1$ colors then there exists a monochromatic connected component consisting of at least $f(d, m) n^{d-m}$ small cubes.

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…

Extending the notion of $r$-(class) regular partitions, we define $(r_{1},...,r_{m})$-class regular partitions. A partition identity is presented and described by making use of the Glaisher correspondence.

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…

Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These…

In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…

In 1918, Hardy and Ramanujan made a breakthrough by developing the circle method to deduce an asymptotic formula for the partition function $p(n)$, which was later refined by Rademacher in 1937 to produce an absolutely convergent series…

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a…

For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some…